Mechanism for reversing hard-wired logic
Yes, of course, it's possible but much more tedious than with programmable logic. It is almost worthwhile to install a micro-controller to manage the inversion mechanism.
Otherwise, with wired logic, you will need three bistable (latching) relays that remain in position successively at each pump start, and from there you can manage the required permutation. When all three relays are on, you perform the forced reset—all this with time delays.