We present a dynamical analysis to infer the structural parameters and properties of the two nearby, compact, high-velocity dispersion galaxies MRK 1216 and NGC 1277. Combining deep Hubble Space Telescope imaging, wide-field integral field unit stellar kinematics, and complementary long-slit spectroscopic data out to three effective radii, we construct orbit-based models to constrain their black hole masses, dark matter content and stellar mass-to-light ratios. We obtain a black hole mass of log(M•/M⊙) = 10.1|$_{-0.2}^{+0.1}$| for NGC 1277 and an upper limit of log(M•/M⊙) = 10.0 for MRK 1216, within 99.7 per cent (3σ) confidence. The stellar mass-to-light ratios span a range of ϒV = |$6.5_{-1.5}^{+1.5}$| in NGC 1277 and ϒH = |$1.8_{-0.8}^{+0.5}$| in MRK 1216 and are in good agreement with single stellar population models of a single power-law Salpeter initial mass function. Even though our models do not place strong constraints on the dark halo parameters, they suggest that dark matter is a necessary ingredient in MRK 1216, with a dark matter contribution of |$22^{+30}_{-20}$| per cent to the total mass budget within one effective radius. NGC 1277, on the other hand, can be reproduced without the need for a dark halo, and a maximal dark matter fraction of 13 per cent within the same radial extent. In addition, we investigate the orbital structures of both galaxies, which are rotationally supported and consistent with photometric multi-Sérsic decompositions, indicating that these compact objects do not host classical, non-rotating bulges formed during recent (z ≤ 2) dissipative events or through violent relaxation. Finally, both MRK 1216 and NGC 1277 are anisotropic, with a global anisotropy parameter δ of 0.33 and 0.58, respectively. While MRK 1216 follows the trend of fast-rotating, oblate galaxies with a flattened velocity dispersion tensor in the meridional plane of the order of βz ∼ δ, NGC 1277 is highly tangentially anisotropic and seems to belong kinematically to a distinct class of objects.