Are you familiar with a mobius strip? A ribbon with a twist in it produces a one-sided band. A klein bottle is essentially like a levelling-up of that by a dimension. It is a 4D surface which is, just like the mobius strip, single-sided. If you were 4th dimensional, you could trade from the 'outside' to the 'inside' and back without crossing any boundary. Of course, the klein bottles you can buy or see renderings of are 3D embeddings of this, so they have to include a bit of self-intersection that wouldn't actually occur in the real 4D version.