I've had a similar experience looking for my wife in a store (she'd be behind a shelving unit as I looked down the aisles). We weren't constrained by mirroring of actions, but finding her was still more difficult (and frustrating) than I thought it would be.

Even if the twins / parent and child / spouses were not obligated to move symmetrically, it's still POSSIBLE for them to continuously not see each other, especially if the barrier is large enough.

Suppose that after I didn't see my wife in any of the aisles as I walked across one end of the store, I then walked down an aisle (doesn't matter which) to look down the other side, but at the same time my wife walked up another aisle. When each of us reached the opposite end, we would not see the other one.

In theory, the search could go on indefinitely (if the store was open 24 hours), and at Wal-Mart (with horizontal as well as vertical aisles) I've sometimes spent 30 minutes looking.

As with the Buridan's donkey paradox, we could theoretically die of starvation on our Kafkaesque quests. (At Wal-Mart, my wife eventually paged me over the store's speakers; in a smaller store like a Dollar General, one of us would usually shout to the other one).

It would be interesting to formulate a problem (eliminating kludges like paging and shouting) to determine probabilistically how long it would take a non-constrained pair to find each other. But that's way beyond my mathematical ability.

A social scientist might devise an experiment to see how long it would take before one of the couple would resort to a kludge. This approach might help resolve the problem of determinism.