Reduce inequality does not finish

I am trying to show general cases of symmetry of regions.

However, for the following, Reduce does not finish:

Reduce[ForAll[x, 
    x > 0, ! (RegionMember[ImplicitRegion[(0 < x < 1 || -2 < x < 0 || 2 < x < 3), {x}], {x}] [Xor] 
              RegionMember[ImplicitRegion[(0 < x < 1 || -2 < x < 0 || 2 < x < 3), {x}], {-x}])],
    x, Reals]

OTOH, if I remove any one of the three conditions (from both parts), e.g.:

Reduce[ForAll[x, 
    x > 0, ! (RegionMember[ImplicitRegion[(-2 < x < 0 || 2 < x < 3), {x}], {x}] [Xor] 
              RegionMember[ImplicitRegion[(-2 < x < 0 || 2 < x < 3), {x}], {-x}])],
    x, Reals]

it return False immediately.

What’s happening?

(I know there are other ways to do this, e.g., with FindInstance.)

Thanx.

reg1 = ImplicitRegion[(0 < x < 1 || -2 < x < 0 || 2 < x < 3), {x}]
reg2 = ImplicitRegion[(0 < x < 1 || -2 < x < 0 || 2 < x < 3) /. 
   x -> -x, {x}]
RegionEqual[reg1, reg2]

Reduce[ForAll[x,  x > 0, ! (RegionMember[
   ImplicitRegion[(0 < x < 1 || -2 < x < 0 || 
      2 < x < 3), {x}], {x}] [Xor] 
  RegionMember[
   ImplicitRegion[(0 < x < 1 || -2 < x < 0 || 
      2 < x < 3), {x}], {-x}]) // Simplify], x, Reals]
(*False*)