  In this work a symmetry of universal finite-size scaling functions under a
certain anisotropic scale transformation is postulated. This transformation
connects the properties of a finite two-dimensional system at criticality with
generalized aspect ratio $\rho > 1$ to a system with $\rho < 1$. The symmetry
is formulated within a finite-size scaling theory, and expressions for several
universal amplitude ratios are derived. The predictions are confirmed within
the exactly solvable weakly anisotropic two-dimensional Ising model and are
checked within the two-dimensional dipolar in-plane Ising model using Monte
Carlo simulations. This model shows a strongly anisotropic phase transition
with different correlation length exponents $\nu_{||} \neq \nu_\perp$ parallel
and perpendicular to the spin axis.
