  Let $C$ be a smooth projective curve and $G$ a finite subgroup of
$\mathrm{Aut}(C)^2\rtimes \mathbb Z_2$ whose action is \textit{mixed},
i.e.~there are elements in $G$ exchanging the two isotrivial fibrations of
$C\times C$. Let $G^0\triangleleft G$ be the index two subgroup
$G\cap\mathrm{Aut}(C)^2$. If $G^0$ acts freely, then $X:=(C\times C)/G$ is
smooth and we call it \textit{semi-isogenous mixed surface}. In this paper we
give an algorithm to determine semi-isogenous mixed surfaces with given
geometric genus, irregularity and self-intersection of the canonical class. As
an application we classify irregular semi-isogenous mixed surfaces with $K^2>0$
and geometric genus equal to the irregularity; the regular case is subjected to
some computational restrictions.
  In this way we construct new examples of surfaces of general type with
$\chi=1$. We provide an example of a minimal surface of general type with
$K^2=7$ and $p_g=q=2$.
