According to your image, it seems like the thickened Klein bottle deformation retracts to the front face of it, which in turn is a Klein bottle. If that is the case, then see here. Anyhow, can you define the thickened Klein bottle? My algebraic topology books (Hatcher's and Massey's) doesn't introduce the thickened Klein bottle at all. Is it simply K X [0,1], where K is the Klein bottle?