We analyze the rotational dynamics of spherical colloidal Janus particles made from silica (SiO2) with a hemispherical gold/palladium (Au/Pd) cap. Since the refractive index difference between the surrounding fluid and a two-faced, optically anisotropic Janus microsphere is a function of the particle's orientation, it is possible to observe its rotational dynamics with bright-field optical microscopy. We investigate rotational diffusion and constant rotation of single Janus microspheres which are partially tethered to a solid surface so they are free to rotate but show little or no translational motion. Also, since the metal cap is a powerful catalyst in the breakdown of hydrogen peroxide, H2O2, the particles can be activated chemically. In this case, we analyze the motion of coupled Janus dimers which undergo a stable rotary motion about a mutual center. The analysis of both experimental and simulation data, which are microscopy and computer-generated videos, respectively, is based upon individual particle tracking and differential dynamic microscopy (DDM). DDM, which typically requires ensemble averages to extract meaningful information for colloidal dynamics, can be effective in certain situations for systems consisting of single entities. In particular, when translational motion is suppressed, both rotational diffusion and constant rotation can be probed.