We present general relativistic magnetohydrodynamical simulations of equal-mass spinning black hole binary mergers embedded in a magnetized gas cloud. We focus on the effect of the spin orientation relative to the orbital angular momentum on the flow dynamics, mass accretion rate, and Poynting luminosity. We find that, across the inspiral, the gas accreting onto the individual black holes concentrates into disklike overdensities whose angular momenta are oriented toward the spin axes and that persist until merger. We identify quasiperiodic modulations occurring in the mass accretion rate at the level of ∼1%–20%, evolving in parallel with the gravitational-wave chirp. The similarity between the accretion rate time series and the gravitational strain is a consequence of the interplay between strong, dynamical gravitational fields and magnetic fields in the vicinity of the inspiraling black holes. This result suggests that quasiperiodicity in the premerger accretion rate of massive binaries is not exclusive of environments in which the black holes are embedded in a circumbinary accretion disk and could provide an additional useful signature of electromagnetic emission concurrent to low-frequency gravitational-wave detection.