  We call a projective surface $X$ mixed quasi-\'etale quotient if there exists
a curve $C$ of genus $g(C)\geq 2$ and a finite group $G$ that acts on $C\times
C$ exchanging the factors such that $X=(C\times C)/G$ and the map $C\times C
\rightarrow X$ has finite branch locus. The minimal resolution of its
singularities is called mixed quasi-\'etale surface. We study the mixed
quasi-\'etale surfaces under the assumption that $(C\times C)/G^0$ has only
nodes as singularities, where $G^0\triangleleft G$ is the index two subgroup of
the elements that do not exchange the factors. We classify the minimal regular
surfaces with $p_g=0$ whose canonical model is a mixed quasi-\'etale quotient
as above. All these surfaces are of general type and as an important byproduct,
we provide an example of a numerical Campedelli surface with topological
fundamental group $\bbZ_4$, and we realize 2 new topological types of surfaces
of general type. Three of the families we construct are $\bbQ$-homology
projective planes.
