Assigning Phase Flows to Phase Curves

Given a family of phase curves on the plane, how many different ways can we assign phase flows to it? i.e. how many different ways can we assign orientation to the curves (identifying them as phase flows) so that corresponding vector fields are smooth?

For instance, if we take the family of curves shown on p. 75 of Arnold’s ODE:

There are two ways in which we can assign orientation to the curves:

Notice that once we have fixed the orientation of one curve, the orientations of the other curves are fixed.

Is it possible that the orientation of one curve does not determine the orientation of the remaining curves on the plane?

Yes. Notice the 4 phase spaces below share the same phase curves, but the orientations of their phase flows are different in each case:

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