The outer part is reminiscent of these:
...but these are microfossils and much smaller than what you have. These are Citharina sp. Not sure they are related but I thought it was worth mentioning! Also considering you have found lots of these, at this size and with a larger attachment I doubt this is related, but its something to think about.
Thanks very much to Louise and Ryan for the input, but I don't think we're there yet.
There is more to these fossils than the parallel grooves. They all taper to a blunt tip making a triangle shape. Opposite the grooves, and almost perpendicular to them, there is usually a grain like texture radiating from the tip. I found one today that demonstrates this, as it has light and dark layers (top). Also, there is often a smooth depression in the middle at the broken end.
Do the right and left handed versions mean that one organism had a pair of them? Possibly closing together?
I hadn't been able to find anything like it in books until I went back 200 years to one by James (he named the disease) Parkinson called, Organic Remains of a Former World: Volume 3 (Google books). In there I found a picture (XV, 8) of a shell labelled Perna from northern Italy.
Googling Perna I found a name change to Mulletia mulleti. Can anybody check one of these fossils for similarities to the ones I'm finding? I might be finding broken tips.
Ryan, to answer your question the Gryphaea are mostly whole with fragments of larger ones. Sometimes the grooves in my fossils are fused in rock, so I don’t think they formed recently.
I would say you are almost definately correct- they are the broken remains of the front ends ('beaks') of Bivalve mollusc shells, probably all that remains of the shells after they were rolled and eroded by Jurassic seas. The layers are layers of shell put down by the mollusc as it grew - this structure is fairly distinctive. The grooves are what remains of the hinge teeth which articulated the mollusc's two shells. They could be bits of 'Bakevellia' or 'Parallelodon', but as they are very worn and eroded it's unlikely you can put an exact name to them.
Hope this is helpful,