As far as I understand it a propagator, $D(x-y)$, gives the amplitude for a flow of positive energy-momentum from an earlier event $y$ to a later event $x$.
Addendum: Instead of talking about energy flows I should be more careful and say that the propagator, $D(x-y)$, gives the amplitude that a virtual particle created at event $y$ will be annihilated at a later event $x$ giving up a certain amount of positive energy-momentum to the absorber. One cannot imply that the virtual particle itself ever has a definite amount of energy-momentum.
But surely in order to conserve energy-momentum one then needs to apply the conjugate operator $D^\dagger(x-y)$ giving the amplitude for a balancing flow of negative energy-momentum from event $x$ backwards in time to event $y$?
Addendum: Using more careful language one should say that to conserve energy-momentum one needs to apply $D^\dagger(x-y)$ giving the amplitude that a virtual particle created at event $x$ travels backwards in time to event $y$ giving up a balancing amount of negative energy-momentum to the emitter.
Thus the overall amplitude for the complete propagation process of a particle from emitter at $y$ to absorber at $x$ is $P(x-y)$ given by:
$$P(x-y) = D(x-y) D^\dagger(x-y) = |D(x-y)|^2$$
which is in fact the probability of a particle propagating from $y$ to $x$.
Maybe the requirement of conservation of energy-momentum actually implies the Born rule?
PS Normally energy-momentum is said to be conserved at the vertices of Feynman diagrams. But if one applies conservation of energy-momentum at a vertex where a virtual particle is created then surely one is implying that the virtual particle has a definite energy-momentum which is not allowed?
Addendum 2: Summary in terms of CPT symmetry
In QED the scattering amplitude is expressed in terms of time-ordered Feynman diagrams so that is necessarily time-asymmetric.
But Maxwell's equations are symmetric with respect to time + parity + charge conjugation (CPT).
I think the solution is that when one calculates scattering probabilities using the Born rule one multiplies the QED scattering amplitude by its complex conjugate. That complex conjugate amplitude is made up of time-reversed, parity-reversed, charge-reversed Feynman diagrams. Thus the (real) product is CPT-symmetric as it should be to be consistent with Maxwell's equations.
Is this reasoning correct?