In my native language we have a neither-nor-like structure which can be used either as:
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:
As @Richard points out, putting "neither" before the verb implies that the verb is part of what the "neither" is applying to:
In English, the sequence must be 'neither ... nor ...'. You also do not use an 'and' in the construction:
The second is generally better where the verb ('like') is the same; the first is necessary if the verbs differ:
If the noun phrases ('thing A' in the example) are in fact compound phrases, then the repeated verb might still be beneficial even when it is the same in both parts (though I'm not convinced this example manages to demonstrate the point):
However, at that point, it might be written as: