Minimum critical set size for Sudoku Pairs Define $Sudoku$ $Pair$ as a pair of mutually orthogonal Sudoku Squares. For example:
$$
\begin{bmatrix}
  55 & 18 & 66 & | & 71 & 89 & 32 & | & 93 & 24 & 27 \\
  87 & 34 & 41 & | & 28 & 96 & 53 & | & 69 & 12 & 75 \\
  99 & 22 & 73 & | & 14 & 67 & 45 & | & 56 & 31 & 88 \\
  -- & -- & -- & + & -- & -- & -- & + & -- & -- & -- \\
  78 & 61 & 39 & | & 85 & 52 & 16 & | & 44 & 97 & 23 \\
  26 & 83 & 54 & | & 37 & 48 & 91 & |  & 72 & 65 & 19 \\
  42 & 95 & 17 & | & 63 & 74 & 29 & | & 38 & 86 & 51 \\
  -- & -- & -- & + & -- & -- & -- & + & -- & -- & -- \\
  33 & 57 & 82 & | & 46 & 21 & 68 & | & 15 & 79 & 94 \\
  64 & 49 & 25 & | & 92 & 13 & 77 & | & 81 & 58 & 36 \\
  11 & 76 & 98 & | & 59 & 35 & 84 & | & 27 & 43 & 62 \\
\end{bmatrix}$$
It is now well known that the minimum $critical$ $set$ size for normal Sudoku Squares is $17$, so it takes at least $34$ clues to fix 2 independent Sudoku Squares. 
For $Sudoku$ $Pairs$ we can clearly do better. Here is an example using 15 paired clues for $Sudoku$ $Pair$ above:
$$
\begin{bmatrix}
   . &  . & 66 & | &  . &  . &  . & | &  . &  . & 27 \\
  87 &  . &  . & | &  . &  . &  . & | &  . & 12 &  . \\
   . &  . &  . & | &  . &  . &  . & | &  . &  . & 88 \\
  -- & -- & -- & + & -- & -- & -- & + & -- & -- & -- \\
  78 &  . &  . & | &  . &  . &  . & | &  . &  . &  . \\
   . &  . &  . & | &  . &  . &  . & | &  . &  . &  . \\
   . &  . &  . & | & 63 &  . &  . & | &  . &  . & 51 \\
  -- & -- & -- & + & -- & -- & -- & + & -- & -- & -- \\
   . &  . &  . & | &  . &  . &  . & | &  . & 79 &  . \\
   . & 49 &  . & | &  . &  . & 77 & | &  . & 58 &  . \\
   . &  . &  . & | &  . & 35 &  . & | & 27 &  . & 62 \\
\end{bmatrix}$$
The question naturally arises - just how low can we go? What is the minimum size of a critical set? 
 A: I have discovered since posting this question that the $30$-clue set above is not minimal for this grid, in fact we can remove at least 7 clues, giving a $23$-clue set ($x$ indicates a removed clue):
$$
\begin{bmatrix}
   . &  . & 66 & | &  . &  . &  . & | &  . &  . & x7 \\
  8x &  . &  . & | &  . &  . &  . & | &  . & 12 &  . \\
   . &  . &  . & | &  . &  . &  . & | &  . &  . & 8x \\
  -- & -- & -- & + & -- & -- & -- & + & -- & -- & -- \\
  78 &  . &  . & | &  . &  . &  . & | &  . &  . &  . \\
   . &  . &  . & | &  . &  . &  . & | &  . &  . &  . \\
   . &  . &  . & | & 63 &  . &  . & | &  . &  . & 51 \\
  -- & -- & -- & + & -- & -- & -- & + & -- & -- & -- \\
   . &  . &  . & | &  . &  . &  . & | &  . & 79 &  . \\
   . & 4x &  . & | &  . &  . & 7x & | &  . & x8 &  . \\
   . &  . &  . & | &  . & 35 &  . & | & 27 &  . & 6x \\
\end{bmatrix}$$
The process of testing reductions seems to be especially challenging computationally. The sequence of 7 clue removals required 5 minutes CPU time for the first, ranging up to 75 hours (!) for the 7th.
And there are still 3 potential candidates whose removability remains undetermined, before I can declare the clue-set fully reduced.
