Are you familiar with the construction of a torus by taking a square and "gluing the edges together"? It's effectively the game Astroids, where going off one edge of the screen wraps around to the opposite edge.

If, instead, you twist the edges 180 degrees before gluing, we then get the Klein Bottle. I.e. when you warp from, say, the left edge to the right edge, you also get flipped vertically.

I'm guessing you're already familiar with that, but by a similar construction, we can start with a (hyper-)cube and glue faces together to get higher dimensional analogues.

To my knoweldge the easiest visualization of a non-orientable 3D manifold is just by thinking of a cube that warps around to the opposite face. However, in addition, it also reflects you into your mirror image. This is essentially a Klein bottle in one higher dimension.

It's then not too hard to start grappling with the 4D case and beyond!