"Neither . . . nor" vs. "nor . . . neither"

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.

In English, the sequence must be 'neither ... nor ...'. You also do not use an 'and' in the construction:

I neither like thing A nor like thing B.

I like neither thing A nor thing B.

The second is generally better where the verb ('like') is the same; the first is necessary if the verbs differ:

I neither like thing A nor dislike thing B.

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):

I neither like the carelessness of the average mailing list correspondent nor like the tendency of mailing lists to degenerate into ad hominem attacks.

However, at that point, it might be written as:

I do not like the carelessness of the average mailing list correspondent, nor do I like the tendency of mailing lists to degenerate into ad hominem attacks.