I have never seen this kind of notation on a piano score before? Does anyone know what they are?
I agree with hirschme's comment. Specifically, I think the bass clef is perhaps using some sort of default percussion noteheads for specific staff locations. Note that X noteheads appear at G2, D3, E3, B3, D4, and E4. Forward arrow noteheads appear only at G3, while the weird slanted diamond noteheads appear on F3. A3, C4, and notes above E4 all appear as normal noteheads.
Given the consistency of the non-standard noteheads appearing in specific staff locations, I can only assume this is either some percussion noteheads set to correspond to specific staff locations, or else the typesetter has accidentally turned on some other setting (kind of like shape-note noteheads that some music software can do, except that isn't it here, as these are not consistent with the way shape-note notation works).
In sum, unless you can find some sort of rationale or "key" for these noteheads, my guess is that the typesetter made some sort of mistake in settings and accidentally altered the notehead appearance for certain staff positions in the left hand.