The basic element of the labyrinthine receptor organs that transduces mechanical force to nerve action potentials is the hair cell.  The adequate stimulus for hair cell activation is a force acting parallel to the top of the cell.

There are a number of parallel pathways connecting the three semicircular canals (horizontal, anterior, and posterior) and the two otolith organs (saccule and utricle) in each labyrinth to the six extraocular muscles in each eye.

With the two utricles and saccules oriented at right angles to each other, the direction of linear acceleration is spatially encoded in three dimensions, and the magnitude of the acceleration is encoded by the firing rate of afferents in the vestibular nerve.  As the head rotates, the inertial force of the fluid in the semicircular canals deflects the cilia of hair cells aligned with the canals, modulating the firing of the afferent nerves. Since the three semicircular canals aligned at right angles to each other, rotation in any direction can be resolved.

The two otolith organs sense linear acceleration and its gravitational equivalent, and the three semicircular canals sense rotational movement in space¹. 

•   The utricle and horizontal canal work together to decode horizontal movements.
•   The saccule and the two vertically orientated semicircular canals (anterior and posterior) work together to decode vertical movements.

These systems also work together: for example, for a tilt of the head to one side: the semicircular canals will report transient head rotation, and the otolith organs will signal head tilt.

Redrawn from: Day BL, Fitzpatrick RC. The vestibular system. Curr Biol. 2005;15(15):R583-6.

The semicircular canals and otoliths also provide information for vestibulo-spinal reflexes that keep the head and body upright.  Both semicircular canals and otoliths can detect motion in three dimensions; ie around any or all of the three axes of rotation: pitch (vertical), yaw (horizontal) and roll (torsional), and around any or all of the three axes of translation: fore and aft, side-to-side, and up-and-down¹. 
Normally, the semicircular canal and otolith organs operate as a unit under conditions of free head movement, but in an experimental setup the semicircular canals may be stimulated separately by giving yaw, pitch, roll or linear acceleration:
 

From: Arcoverde E, Duarte R, Barreto R, Magalhaes, J, Bastos C, Ing Ren T,Cavalcanti G. 2014. Enhanced real-time head pose estimation system for mobile device. Integrated Computer Aided Engineering. 21. 281-293. 10.3233/ICA-140462.

Since the semicircular canals are orientated to one another virtually at right angles,  rotation in any direction will be sensed by one or more of the cristae. As in any Cartesian three coordinate system, any direction of rotation can be signaled uniquely by the combined discharges from the three canals³. ( See: Cartesian coordinate system)
 

(vv)Vestibular System-24.mp4(tt)
 

Note that motion can be broken down, or decomposed, into rotational and translational. Each of these is defined by degrees of freedom.

There are 3 rotational degrees of freedom:

1. Yaw
2. Pitch
3. Roll

Rotations are detected by the semicircular canals, although it is noteworthy that rotation does not correspond precisely to canal function (except for horizontal (yaw) rotation).

Rotational motion is where an object spins around an internal axis in a continuous way. An ice-skater can do this by spinning on the spot. The ice-skater gives themselves rotational energy. Since energy is always conserved and a smaller object must spin faster to have the same energy, when the ice-skater moves their arms in towards their body, rotation speed will increase - the spinning will get faster and faster.

Translational motion is motion that involves the sliding of an object in one or more of the three dimensions: x, y or z. 
 

(vv)YawRoll.mp4(tt)

From: Purves D, Augustine GJ, Fitzpatrick D, et al., editors. Neuroscience. 5h edition. Sunderland (MA): Sinauer Associates; 2012. Animation 14.1: The Vestibular System. Available from: https://neuroscience5e.sinauer.com/animations14.01.html
 

(vv)differences-between-translational-rotational-motion.mp4(tt)