encryption – Bitwise operations in FHE

Im reading about FHE and the libraries implementing it (SEAL, HELib).
I saw that SEAL doesn’t support bitwise operations but I wondered if its theoretically feasible.
For example, bitwise-ing XOR an encrypted value with itself, gets us an encrypted 0. Bitwise it with the not of itself to get an encrypted 1. Using shifts with the computed two I would then be able to extract the encrypted number.
Shift right/left can be made using multiplication or division by (powers of) 2.
But XOR-ing is the main problem.
Is it theoretically possible under any FHE/HE scheme? What are the limitations?