Wolfram Language (Mathematica), 28 bytes
GatherBy[#,i=0;Abs[#-++i]&]&
Wolfram Language (Mathematica), 25 bytes
GroupBy[i=0;Abs[#-++i]&]&
Looser IO format. Input [][array]
, and outputs an association whose values are the groups (keyed by their displacements).