MarketPress cart contents come and go

The cart is misbehaving badly every sort of way.

- I put 3 items into my cart, went to checkout, removed one item, and the cart emptied. Later the missing items reappeared.
- I went to the checkout page with some items and decided to continue shopping, as one might, and now the sidebar cart is empty. When I go back to the checkout, they reappear.
- I put one item into my cart and go to checkout to find some other item there.

We tested the cart earlier and it was fine. The client is paid in full. And now this. Please see what's the matter?