Relativistic wave equations for spin s particles (-i β \cdot partial + m) ψ = 0 transforming under a representation Λ mapsto T(Λ) of SL(2, C), can have structural properties that make them physically equivalent to similar wave equations. This equivalence extends to all the standard external field interactions. There are T(Λ)'s which cannot lead to theories different from simpler ones. When the order of the minimal polynomial of β0 is restricted, many more representations lead to equivalence with simpler wave equations. This study investigates the general conditions under which these equivalences are manifested.