point of view of obtaining physical results, one may wish to convert results in a given gauge to a different one [@BMMS; @MMB], to compare results obtained in two different gauges, or to construct a fully gauge invariant formalism. To this end one needs to know the gauge transformation rules and the rules for gauge invariance, either [*up to*]{} order $n$ [@BMMS] or [*at*]{} order $n$ [*only*]{}, as in [@CPM]. The situation is going to be more complicated in the case of two parameters, as we shall see.[^1]

In this paper we keep in mind the above practical examples, but we do not make any specific assumption on the background spacetime and the two–parameter family it belongs to. As in [@BMMS; @SB; @BS], we do not even need to assume that the background is a solution of Einstein’s field equations: the formalism is completely general and can be applied to any spacetime theory. We analyse the gauge dependence of perturbations in the case when they depend on two parameters, $\lambda$ and $\Omega$, derive explicit gauge transformation rules up to fourth order, i.e. including any term $\lambda^k\Omega^{k'}$ with $k+k'\leq 4$, and define gauge invariance. This choice of keeping fixed the total 