Thank you for your response.
However, the problem exists even when I turn the template off.
So the template provider is suggesting the problem lies elsewhere.
On the other hand, it seems to have begun occurring at the same time as changes made to jQuery settings to correct other issues with the template.
To re-state what is happening, with a little more detail. If you one product to cart as unregistered shopper, no problem. If you checkout, it asks you to register, takes you through, everything works a treat. But if instead of checking out, you keep shopping and adding more products to the cart, now the cart will only show the first product that was put in. If you then nevertheless go ahead and log in, everything re-appears in the cart, and everything works.
But now also: if you decide not to checkout, and instead delete all the products from your cart, so that cart is empty, everything is fine. But when you log out, the cart now still shows the first product added. Proceed to checkout, it's not there. Go shopping again, add another product, go to cart: still the same first product is there and cannot be budged.
I will go back talk to the template developer some more. Thanks for your inputs.