A pseudoclassical model is proposed for the description of planar P, T-invariant massive fermions. The quantization of the model leads to the (2+1)-dimensional P, T-invariant fermion model used recently in P, T-conserving theories of high- Tc superconductors. The rich symmetry of the quantum model is elucidated through the analysis of the canonical structure of its pseudoclassical counterpart. We show that both the quantum P, T-invariant planar massive fermion model and the proposed pseudoclassical model - for a particular choice of the parameter appearing in the Lagrangian - have a U(1,1) dynamical symmetry as well as an N = 3 supersymmetry. The hidden supersymmetry leads to a nonstandard superextension of the (2+1)-dimensional Poincaré group. In the quantum theory the one-particle states provide an irreducible representation of the extended supergroup labelled by the zero eigenvalue of the superspin. We discuss the gauge modification of the pseudoclassical model and compare our results with those obtained from the standard pseudoclassical model for massive planar fermions.