  Pull-back transformations between Heun and Gauss hypergeometric equations
give useful expressions of Heun functions in terms of better understood
hypergeometric functions. This article classifies, up to Mobius automorphisms,
the coverings P1-to-P1 that yield pull-back transformations from hypergeometric
to Heun equations with at least one free parameter (excluding the cases when
the involved hypergeometric equation has cyclic or dihedral monodromy). In all,
61 parametric hypergeometric-to-Heun transformations are found, of maximal
degree 12. Among them, 28 pull-backs are compositions of smaller degree
transformations between hypergeometric and Heun functions. The 61
transformations are realized by 48 different Belyi coverings (though 2
coverings should be counted twice as their moduli field is quadratic). The same
Belyi coverings appear in several other contexts. For example, 38 of the
coverings appear in Herfutner's list of elliptic surfaces over P1 with four
singular fibers, as their j-invariants. In passing, we demonstrate an elegant
way to show that there are no coverings P1-to-P1 with some branching patterns.
