One Knot Split Over Rules

A topological instance of two ruled surfaces is shown, as produced from splitting a torus in two, by way of one knot-band splitting.  The boundary edge is a trefoil and the enclosed space is a knot-torus. 

The idea is essentially that different rulers would mate such that the offspring inherits from both; each set of rulings in their own right is a mobius band or a torus, respectively, one knot like the other, if extended back to parental form. 

Somehow in my inspiration for this object, to quote a famous psychologist:  the desire of the subject is the desire of the other.   I desired one not like another, yet desire is overruled.