Orthogonal Orientation
MATL, 10 6 bytes

4 bytes saved thanks to Martin!

19\qX!


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19\            % implicitly take input (a character) and compute mod-19 of its ASCII code
   q           % subtract 1. Gives 17, 2, 3, 4 for the characters '^&lt;v&gt;' respectively.
               % These numbers correspond to 1, 2, 3, 4 modulo 4, and so are the numbers
               % of 90-degree rotations required by each character
    X!         % implicitly take input (string). Rotate the computed number of times
               % in steps of 90 degrees. Implicitly display




Old version, without modulo operations: 10 bytes

'^&lt;v&gt;'=fX!


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'^&lt;v&gt;'         % push string
      =        % implicitly take input (a char) and test for equality
       f       % find index of matching character
        X!     % implicitly take input (string). Rotate that number of times
               % in steps of 90 degrees. Implicitly display