Slither Like a Snake
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UÐeẎṙṁ⁸UÐe

A dyadic Link accepting the marix on the left and the rotation integer on the right (uses the reverse meaning of positive / negative)
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UÐeẎṙṁ⁸UÐe - Link: matrix of integers, M; integer, R
 Ðe        - apply to even indices of M:
U          -   reverse each
   Ẏ       - tighten
    ṙ      - rotate left by R
     ṁ     - mould like:
      ⁸    -   chain's left argument, M
        Ðe - apply to even indices:
       U   -   reverse each