 Double covers of the figure-eight space

There are three connected two-sheeted covering spaces of the wedge of two circles, distinguished by the three nonzero homomorphisms from its fundamental group to $\mathbb{Z}/2\mathbb{Z}$. If exactly one circle exchanges sheets, the covering graph has two parallel joining edges and one loop at each vertex. If both circles exchange sheets, it has four parallel joining edges. These give two homeomorphism types of total spaces even though they give three inequivalent covers of the fixed base.