It's an easy proposition that a homotopically nontrivial map between spheres (with the standard metric) has Lipschitz constant at least 1. I prove the following analogous theorem: a map between standard spheres which is not homotopic to a suspension has Lipschitz constant at least 2, with equality exactly when the map is a Hopf fibration. To prove this, I develop a generalization of the Hopf invariant, which distinguishes suspensions from non-suspensions via the topology of a pair of preimages.