In this paper we formulate the BRST quantization of the spinless relativistic point particle without restricting the parameter space to one fundamental domain of the modular group. Modular invariance is instead enforced by a constraint on the physical subspace of the Hilbert space. The second-quantized theory has then a manifest IOSp( D, 2|2) invariance. Studying this symmetry we find that a special element of the SO( D, 2) subgroup of IOSp( D, 2|2) can be identified which induces the PCT transformation on the physical subspace. Using this, the PCT theorem is derived as a simple consequence of the IOSp( D, 2|2) invariance, thus confirming the relevant role of this graded symmetry. The subgroup SO( D, 2) seems to play a central role also in another respect: by requiring covariance of the BRST quantization with respect to it, and not just invariance with respect to the Lorentz group SO( D - 1,1) the overall IOSp( D, 2|2) symmetry is generated.