Specular reflections of relativistic laser pulses from an overdense plasma mirror (PM) were studied experimentally. The pointing stability of the PM and reflectance of the input laser were characterized. The solid material used for the PM was a VHS tape. This study was done for the magnetic and plastic sides of the VHS tape, and for input light of both s and p-polarizations. The laser pulse fluence was varied by changing the focus position relative to the tape surface, which changed the spot size at the tape. The pointing fluctuations of the reflected pulses caused by the PM were 1 mrad. A peak reflectance of 82% was obtained from the plastic surface of the VHS tape when focusing s-polarized light 4 mm from the tape surface (the wavefront quality was confirmed to be conserved). An analytic model was developed to understand the physics of the interaction for each tape material and polarization. Fitting of our model parameters to the experimental results allowed an estimate of the key plasma parameters such as plasma expansion velocity, ionization intensity, and fraction of absorbed laser energy.