We propose a new axiomatisation of the alpha-equivalence relation for nominal terms, based on a primitive notion of fixed-point constraint. We show that the standard freshness relation between atoms and terms can be derived from the more primitive notion of permutation fixed-point, and use this result to prove the correctness of the new alpha-equivalence axiomatisation. This gives rise to a new notion of nominal unification, where solutions for unification problems are pairs of a fixed-point context and a substitution. This notion of nominal unifier behaves better than the standard notion based on freshness contexts: nominal unification remains finitary in the presence of equational theories such as commutativity, whereas it becomes infinitary when unifiers are expressed using freshness contexts.