Sorting for search by monotomy
Jelly, 11 9 bytes

JBÇṙ€1ỤỤịṢ


Try it online! Explanation:

J           Create a range 1-(length of input array).
 B          Convert all the elements to binary.
  ṙ€1       Rotate each element left by 1.
     ỤỤ     Find each element's position within the sorted list.
       ịṢ   Apply that permutation to the sort of the input.