We show how the existence of an a-theorem imposes constraints on the beta-functions which may be expressed in terms of renormalisation-scheme-invariant combinations of the beta-function coefficients. This motivates a general discussion of scheme-invariance; we show that the structure of scheme transformations implies significantly more scheme-invariants than might be naively expected. Finally we show that the language of Hopf algebras provides a convenient means to describe the requirements of scheme invariance.