Is there any known pair of objects tidally locked with each other? I know the moon is tidally locked with the earth (e.g. one side always facing earth), but I don't know of any pair of objects that are both tidally locked with each other. For example, the planet would be tidally locked with the moon, and the moon would be tidally locked with the planet.
Pluto and its largest moon Charon are tidally locked to each other.
Charon and Pluto revolve about each other every 6.387 days. The two objects are both gravitationally locked to the other, so each keeps the same face towards the other. This is a mutual case of tidal locking . . .
Because of Charon's large size compared to Pluto, and because its barycenter lies above the primary's surface, some astronomers have referred to the Pluto/Charon system as a dwarf double planet.
The system is also unusual among planetary systems in that each is tidally locked to the other: Charon always presents the same face to Pluto, and Pluto always presents the same face to Charon: from any position on either body, the other is always at the same position in the sky, or always obscured.(110)
Consider the case of similar-mass, close binary systems. If the orbital periods are less than a few days they are expected to become tidally locked on a timescale much shorter than the stellar lifetimes.
In many cases the synchronisation can be established by comparing the rotation periods of the stars using modulation by starspots with the orbital period from eclipses e.g. http://adsabs.harvard.edu/abs/1983A%26A...123..316E
In other cases you can find the projected equatorial velocities of eclipsing binary star components and show that these are consistent with the orbital period, the measured radii and the orbital inclination e.g. http://adsabs.harvard.edu/abs/2009ApJ...691.1400M ; http://adsabs.harvard.edu/abs/2002ApJ...567.1140T