In fact, there are two sets of Lagrangian triples right here in our own back yard: Saturn's moons Tethys-Telesto-Calypso and Dione-Helene-Polydeuces.
(If you have not previously read my blog on Orbital Libration Points, some of the following may not make a lot of sense.)
Lagrangian moons and planets are able to share an orbit in a stable configuration, located 60° apart. While it is inviting to imagine a system of six worlds equally spaced around a given orbit—and thus all in one another’s L4 and L5 points—studies have shown that without some other strongly stabilizing influence, such a system would not remain stable, especially for highly elliptical orbits. 
While all that follows is applicable to moons, for simplicity, I'll only discuss Lagrangian planets.
There are a number of possible configurations of planet types as Lagrangian sets, such as an ice giant with two terrestrials at its L4 and L5; a gas giant with two ice giants each leading and trailing in its orbit; a Tellurean with two dwarf-planet companions, etc. Sean Raymond at Planetplanet.net  has written a truly fascinating series of blog posts in which he explores the various possibilities.
Lagrangian pairs are not double planets (see below), as they do not orbit a barycenter common only to the two of them, but each independently orbits the central star(s), both on the same orbit. Also, double planets orbit their common barycenter fairly close to each other—typically within a few tens of their own radii. Lagrangian pairs are much farther apart: indeed, since they are separated by 60° on the same orbit, they are as far from one another as each is from the center of the star system. (See illustration above). Inhabitants of small enough Lagrangian-pair planets might not even be aware of the other planet until an indigenous culture on one or the other invents telescopy. Asteroid 2010 TK7, which orbits at Earth's L4 point, was only discovered in 2010, and then only by the space-based WISE telescope.
This is compounded by the distances involved. As it turns out, large enough objects orbiting at Earth's L4 and L5 points would be visible from Earth only just before sunrise and/or just after sunset, in much the same way Venus is. As shown in the diagram below, those areas of Earth's surface in the shaded area would see the companion at L5 as a "morning star", rising before the Sun, and fading from view as the brightness of the daylight sky drowned it out of view. If the object were very large (see below) and had a high albedo, it might be faintly visible in a clear daylight sky. Also, the object would become visible to about half of the daylight side of the Earth during total solar eclipses, when the Sun's light is blocked. If the companion were at L4, visibility would be as an "evening star", descending below the horizon shortly after sunset.)
Big Enough To See?
Using the equation for the small-angle approximation:
Distance between Earth and Danu: 1.496E+08 km
Actual diameter of Earth: 12742 km
In fact, any object at Earth's L4 or L5 point would have to be the size of the Sun to appear as large in the sky as the full Moon, because that object would be as far from the Earth as is the Sun.
If such a twin had existed in Earth's orbit, it would surely have been noted by the ancients (who did observe the sky during total solar eclipses), and its special nature would have been quickly realized: though it would only appear during an eclipse, its orientation with respect to the Sun in the sky would never change. Once it came under observation using space-based telescopes, it would always be lit as a waxing crescent, though its brightness would likely change due to its axial rotation, differing cloud patterns, and whether land or ocean was visible in the daylit portion (assuming it had oceans).
One distinction that would certainly make sense is that both bodies orbit a common barycenter; this at least distinguishes them from co-orbital bodies such as Lagrangian pairs or orbit-exchanging pairs such as Saturn’s moons Epimetheus and Janus.
Some suggested criteria for double-planets include:
- Both bodies must qualify individually as planets, per the 2006 IAU definition;
- Both orbit a common center-of-gravity (barycenter), which itself orbits their host star(s);
- The barycenter of their combined orbits must lie outside the radius of both bodies;
- The ratio of their masses must approach a value of 1.0.
It is possible to determine all of these criteria. Let's examine how.
I have described calculating the barycenter of a two-body system in this previous blog, but let's revisit the equation here:
The Earth-Moon Example
Using the following values:
Mass of the Earth = 1.0
Mass of the Moon = 0.0123 Earth masses
Average separation between Earth and Moon = 60.336 Earth radii
Variation In The Barycenter
It can happen that the barycenter migrates between lesser-than and greater-than the radius of the primary. Applying the following relation can reveal if this is the case:
rP = the average distance from the primary to the barycenter;
Rp = the radius of the primary (in the same units as rP);
e = the eccentricity of their orbits
Applying the Earth-Moon values:
rP = 0.73312 Earth radii
Rp = 1.0 Earth radius
e = 0.0549
Maximum Possible Separation
The following will be true:
- The more massive of the pair is the primary;
- They will orbit a barycenter which, itself, will orbit the central star(s);
- The barycenter of the double-planet system will lie outside the radius of either body;
- Their orbital eccentricities around the barycenter will be identical;
- They may be closely or widely spaced, limited by the calculation below
The maximum possible separation can be calculated by the equation:
Note that the masses of the two planets must be expressed in units of the mass of the star, which itself is expressed in terms of the Sun's mass.
For a double-planet system of two Earth-mass bodies at 1.0 AU from the Sun:
At a distance of 902.8 million meters, a twin Earth would subtend 0.81˚ in the sky (about 1.6 times the size of the full Moon); it would orbit ~2.35 times farther away than the Moon orbits the Earth, and have an orbital period of ~70.132 days.
The effect such a body would have on the orbit of the Moon involves rigorous mathematics too complex to discuss here.
Radius of the Earth: 6731 km (12742 km in diameter)
Distance to twin: 6731 ⨉ 2.44 = 15545.24 km
Orbital Periods for Double-Planets
Bear in mind that all of the units must be in Earth units for the equation to give results in Earth days. Thus, MP must be the actual mass of the planet divided by Earth’s actual mass; MS must be the actual mass of the secondary divided by Earth’s actual mass; and RS must be the actual radius of the secondary’s orbit divided by the actual radius of the Earth.
Calculating Orbital Period Without Converting To Earth Units
If you don't wish to go to the effort to convert everything to Earth-based units, the following equation uses absolute units for the elements of the system. The same equation can be used in any situation where two bodies are in orbit around one another:
Orbital Speed for Double-Planets
As the website MathIsFun.com states: “Rather strangely, the perimeter of an ellipse is very difficult to calculate!”  They list several approximations, but I’ve chosen to go with the first one they list, which has as its primary limitation the requirement that the semi-minor axis of the ellipse be no less than one-third the length of the semi-major axis.
Since a 3:1 ratio between the semi-major and semi-minor axes results in an eccentricity of 0.94281, and that is an extremely high eccentricity for an elliptical orbit (the highest eccentricity of any known moon in the Solar System is that of Nereid, at 0.75 ), I consider that the inaccuracies of using this method are acceptable. (If for some reason you have a situation where such an insanely high eccentricity exists in your double-planet or planet-moon pairing, please see the mathisfun.com site for more accurate approximations.)
The following is an equation for calculating the approximate perimeter of an ellipse:
The next equation is:
a = 3.844E+5 kilometers
b = 3.838E+5 kilometers (calculated from a and the eccentricity of its orbit);
P = 2.361E+6 seconds
The "Tug-of-War" Ratio
The result is interpreted as follows: the larger the number, the greater the ratio of the gravitational pull of the primary on the secondary as compared to the pull of the star. The magnitude of FT, then, determines whether or not the pair are a double planet.
Below is a table listing the tug-of-war values for various planet-moon pairs:
In Asimov’s words the Moon is, “…neither a true satellite of the Earth nor a captured one, but is a planet in its own right, moving about the Sun in careful step with the Earth.” 
Of course, by modern IAU standards, the Moon would have to be (ahem) called a dwarf planet….
Asimov’s argument is further strengthened by the fact that at no point in its journey around the Sun does the Moon ever travel “backward” relative to the Sun: "It is always ‘falling toward’ the Sun. All the other satellites, without exception, ‘fall away’ from the Sun through part of their orbits, caught as they are by the superior pull of their primary planets – but not the Moon.” 
In other words, all of the other satellites in the Solar System have loops in their orbit which, when viewed from above the plane of the Solar System, means that for part of their orbits they travel in the opposite direction of their primary’s orbital motion. The Moon’s orbit does not have any such loops, because the speed at which the pair are orbiting the Sun is faster than any "backward" velocity the Moon has in its orbit when it is between the Earth and the Sun.
A very helpful explanation of this is given by Helmer Aslaksen at his website Heavenly Mathematics & Cultural Astronomy .
The illustration on the left above was taken from The Book of Knowledge Vol 1, by Arthur Mee and Holland Thompson, Ph.D., published in 1912. 
I suggest the following:
- If the barycenter of a double-body system lies within the body of the primary; and,
- If their Tug-of-War value < 1.0, meaning that the orbit of the secondary does not "loop back",
If, on the other hand:
- The barycenter of a double-body system lies outside the body of the primary; and,
- If their Tug-of-War value < 1.0, meaning that the orbit of the secondary does not "loop back",
Finally, regardless of where the barycenter lies, if their Tug-of-War value ≥ 1.0, meaning that the orbit of the secondary does "loop back", then the pair are not a double-planet, but rather a planet-moon pair.
Therefore, the Earth-Moon system is a close-bound double-planet system.
The only other significant example in the Solar System is the Pluto-Charon system. Ignoring for the moment the fact that Pluto is not a "full-fledged" planet, the following are true:
- Pluto's radius is 1188.3 km;
- The barycenter of the Pluto-Charon system is 2130.22 km from the center of Pluto, outside the physical body of Pluto;
- The Tug-of-War Ratio for the Pluto-Charon system is:
Therefore, by the above argument Pluto-Charon are neither a close- nor a loose-bound double-planet system, but a planet-moon system. (However, we might make an argument for them being a loose-bound double-dwarf-planet system....)
6. Asimov, Isaac (1976). Asimov on Astronomy. Coronet Books. p125–139. ISBN 0-340-20015-4.
7. en.wikipedia.org/wiki/Double_planet - cite_note-Asimov-5
8. en.wikipedia.org/wiki/Double_planet - cite_note-PoV-7
10. Mee, Arthur, and Holland Thompson, Ph.D., The Book of Knowledge Vol 1 (New York, NY: The Grolier Society, 1912), 147.