Bivalves are very common along the London clay sites, they come in a very large variety of species with distinctive diagnostic properties which can be useful for identification of which species it is.
When fresh out of the London Clay they will still have their original white shell, but this is quite fragile and is not often found as the tides can quickly wear down this fragile structure.
Their core is usually preserved in pyrite which will need treatment to prevent decay if you wish to keep the fossil for getting destroyed.
They can be found most commonly within pyrite shingle patches along the beach and occasionally found within the relatively large nodules.
On the right is a diagram for the different parts of the bivalves and the terminology. Below are links to the different species of bivalves.
Image accreditation:
https://wserv3.esc.cam.ac.uk/p1acollections/exhibits/show/type-fossils/item/85