  We propose a systematic study of transformations of $A$-hypergeometric
functions. Our approach is to apply changes of variables corresponding to
automorphisms of toric rings, to Euler-type integral representations of
$A$-hypergeometric functions. We show that all linear $A$-hypergeometric
transformations arise from symmetries of the corresponding polytope. As an
application of the techniques developed here, we show that the Appell function
$F_4$ does not admit a certain kind of Euler-type integral representation.
