It occurred to me that a Mobius strip twists through space. However the depicted Klein bottles (except maybe the Lawson one) seem to discard this twist. I've created my own Klein 'strip' that derives directly from a Mobius strip.
You can find images at:
The first one shows the Klein Strip in the classic mobius view.
The second one is an animation which shows how it directly derives from a Mobius strip.
The third one is an animation that depicts how the rotation steps along the Strip.
I hope that these will qualify as Klein shapes especially as they derive directly from a Mobius.