Strange behavior with Inequality and pattern matching

The issue is pre-evaluation of the pattern. For the ones that evaluate to False:

Inequality[___]
(* True *)

and

Inequality[1, _, x, LessEqual, 2]
(* Inequality[1, _, x] && x <= 2 *)

Neither of those evaluated forms will match

Inequality[1, Less, x, LessEqual, 2]

To fix this, merely add HoldPattern. For instance,

MatchQ[Inequality[1, Less, x, LessEqual, 2], HoldPattern[Inequality[___]]]
(* True *)

Inactivate is helpful in analyzing this problem.

Inactivate[
  Column[
    {MatchQ[Inequality[1, Less, x, LessEqual, 2], _Inequality],
     MatchQ[Inequality[1, Less, x, LessEqual, 2], Inequality[___]],
     MatchQ[
       Inequality[1, Less, x, LessEqual, 2], 
       Inequality[1, Less, x, LessEqual, 2]],
     MatchQ[
       Inequality[1, Less, x, LessEqual, 2], 
       Inequality[_, Less, _, LessEqual, _]],
     MatchQ[
       Inequality[1, Less, x, LessEqual, 2], 
       Inequality[1, _, x, LessEqual, 2]]}],
 MatchQ]

The above results show clearly why the second and last forms evaluate to False. They also suggests a work-around.

Inactivate[
  MatchQ[
    Inequality[1, Less, x, LessEqual, 2], 
    Inequality[1, Less, x, LessEqual, 2]], 
  Inequality]

True

Inactivate[
  MatchQ[
    Inequality[1, Less, x, LessEqual, 2], 
    Inequality[1, Less, _, LessEqual, 2]], 
  Inequality]

True