In my native language we have a neither-nor-like structure which can be used either
I neither like thing A and nor thing B.
I nor like thing A and neither thing B.
Is it possible to do that in English?
No, “neither” must always come first in this construction.
Also it generally comes after the verb in the example you give, not before it:
I like neither thing A nor thing B.
As @Richard points out, putting “neither” before the verb implies that the verb is part of what the “neither” is applying to:
I neither like nor detest thing A.