hi Saxomonica
When my reeds aren’t in use, i leave the tips of 4 reeds standing in half an inch of mouth wash in a plastic jar with the lid on. The reason for the lid is in case the jar gets knocked over.
I do rotate the reeds and have a different pen mark on each reed, and because the tips are left standing in liquid, that gives the reeds a chance to return back to their natural physical state.
When you play them you put them under a lot of stress, often bending them slightly out of shape and a reed will try to spring back to its original structure when resting. They don’t behave like metal which when bent out shape stays out of shape.
If i’m going out to play my sax somewhere else, and it’s not far away, then i just leave the reed on the mouthpiece, then when i get home i just put it back in the jar. Reeds stay damp for quite a long period, depending on the surrounding temperatures, they soon recover in liquids.