We develop the symmetry classification of superconducting gap functions in electron bands that do not transform like the pure spin-1/2 states under the crystal point group operations. Generalizing the commonly used Ueda-Rice prescription, we define the Bloch bases in the presence of spin-orbit coupling in a way which satisfies all symmetry and continuity requirements, and use these bases to construct general multiband pairing Hamiltonians. For single-band pairing, several exceptional cases are identified in which the triplet superconducting gap function does not transform under the point-group operations as a pseudovector, with a significant impact on the nodal structure. Other consequences of a non-pseudospin character of the bands include such exotic features as odd-parity singlet and even-parity triplet interband gaps, line nodes in the conventional (s-wave) channel, etc.