  A mixed quasi-\'etale quotient is the quotient of the product of a curve of
genus at least 2 with itself by the action of a group which exchanges the two
factors and acts freely out of a finite subset. A mixed quasi-\'etale surface
is the minimal resolution of its singularities.
  We produce an algorithm computing all mixed quasi-\'etale surfaces with given
geometric genus, irregularity, and self-intersection of the canonical class. We
prove that all irregular mixed quasi-\'etale surfaces of general type are
minimal.
  As application, we classify all irregular mixed quasi \'etale surfaces of
general type with genus equal to the irregularity, and all the regular ones
with K^2>0, thus constructing new examples of surfaces of general type with
\chi=1. We mention the first example of a minimal surface of general type with
p_g=q=1 and Albanese fibre of genus bigger than K^2.
