You need to re-apply the master to the pages you move around. It doesn't automatically re-generate when you change the order so the left master might now be on the right pages and vice-versa.
Re-apply the masters by dragging them down onto the spreads.
BAD advice, Rik.
Reapplying a master page reasserts all the master objects behind any that were previously overridden. The same thing happens when you have facing pages and add or remove an odd number of pages -- any pages that switch sides get rebuilt with new unreleased master frames for frames that were manually overridden. Text frames that were released automatically by placing text into them without a manual override first should swap positions, so it sounds like the OP is releasing the master frames before adding text.
One way to minimize the impact would be to enable Layout Adjustment before adding the pages, as long as the frames are snapped to guides. You'll still get the second frame behind, but if it has no stroke, fill or content it may not be a problem. Onthe other hand, if youare releasing master frames before entering text, one has to ask why they are on the master page in the first place.
Your solution seems to resolve the problem.