Ophthalmic properties expressed as functions of dioptric power cannot depend on the particular spherocylindrical form (positive or negative cylinder) chosen to represent the power: they are necessarily invariant under spherocylindrical transposition. This condition of invariance places restrictions on the mathematical form that valid ophthalmic functions can assume. Tests are presented for checking the validity of proposed ophthalmic functions and properties. Examples from the literature are examined, including Keating's concept of torsional power and Peters' graphs of expected unaided visual acuity vs. ametropia. The former satisfies the condition of invariance but the latter are shown to violate the condition for ametropias which are close to spherical. The analysis shows partly how the graphs need to be refined. Invariance under spherocylindrical transposition can assist the researcher in developing new concepts and relationships that depend on power expressed in terms of sphere, cylinder, and axis.