  We demonstrate the strong coupling of both magnons and phonons to terahertz
(THz) frequency electromagnetic (EM) waves confined to a photonic crystal (PhC)
cavity. Our cavity consists of a two-dimensional array of air-holes cut into a
hybrid slab of ferroelectric lithium niobate (LiNbO$_3$) and erbium
orthoferrite (ErFeO$_3$), a canted antiferromagnetic crystal. The phonons in
LiNbO$_3$ and the magnons in ErFeO$_3$ are strongly coupled to the electric and
magnetic field components of the confined EM wave, respectively. This leads to
the formation of new cavity magnon-phonon-polariton modes, which we
experimentally observe as a normal-mode splitting in the frequency spectrum and
an avoided crossing in the temperature-frequency plot. The cavity also has a
mode volume of $V=3.4\times10^{-3}\lambda^3\simeq0.5(\lambda/n)^3$ $\mu$m$^3$
and can achieve a Q-factor as high as 1000. These factors facilitate the
pursuit of the fields of THz cavity spintronics and quantum electrodynamics.
