Eq. (\[coordtransf\]). Here $\xi^\mu_{(1,0)}$, $\xi^\mu_{(0,1)}$, $\xi^\mu_{(2,0)}$, $\xi^\mu_{(1,1)}$, and $\xi^\mu_{(0,2)}$ are independent vector fields and $(\epsilon_0,\epsilon_1)$ are any two real numbers satisfying $\epsilon_0+\epsilon_1=1$. Coupling terms like the $\lambda\Omega$ in (\[coord2\]) are the expected new features of the two–parameter case, cf. [@BMMS; @SB]. Our main results are the explicit transformation rules for the perturbations of a tensor field $T$ and the conditions for the gauge invariance of these perturbations.

The paper is organized as follows: in Section \[taylorexp\] we develop the necessary mathematical tools, deriving Taylor expansion formulae for two–parameter groups of diffeomorphisms and for general two–parameter families of diffeomorphisms. In Section \[gauges\] we set up an appropriate geometrical description of the gauge dependence of perturbations in the specific case of two–parameter families of spacetimes. In Section \[gaugeiandt\] we apply the tools developed in Section \[taylorexp\] to the framework introduced in Section \[gauges\], in order to define gauge invariance and formulas for gauge transformations, up to fourth order in the two–parameter perturbative expansion. Section \[conclusions\] is devoted to the conclusions. We follow the notation used previously in [@BMMS; @SB; @BS] for the case of one parameter perturbations.

Taylor expansion of tensor fields {#taylorexp}
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In order to consider the issues of gauge 