A compact Klein surface X is a compact surface with a dianalytic structure. Such a surface is said to be q-hyperelliptic if it admits an involution \phi , that is an order two automorphism, such that X/ < \phi > has algebraic genus q. A Klein surface of genus 1 and k boundary components is a k-bordered torus.
By means of NEC groups, q-hyperelliptic k-bordered tori are studied and a geometrical description of their associated Teichmüller spaces is given.