mjhjmtu

Here is a list of algebraicly identical equations, and I like to delete all the duplicate equations from this list and get only one equation. I tried the following:

Clear[ss];
listEqs = 
 x + y == x^2 + z,
 y + z == x^2 + z,
 x + y == z + x^2,
 y + z == z + x^2,
 x^2 + z == x + y,
 z + x^2 == x + y,
 x^2 + z == y + z,
 z + x^2 == y + x
 ;

DeleteDuplicates[listEqs]

I got:

 x + y == x^2 + z,
 y + z == x^2 + z, 
 x^2 + z == x + y,
 x^2 + z == y + z

Can someone tell me how to reduce the list to only one equation?

EDIT 1

I noticed a misunderstanding of my question. Here is a second list:

listEqs = 
 y + z == x^2 + z,
 a + b == k + g, 
 x + y == x^2 + z, 
 b + a == g + k,
 x + y == z + x^2,
 a + b == g + k,
 y + z == z + x^2,
 x^2 + z == x + y,
 z + x^2 == x + y,
 x^2 + z == y + z,
 z + x^2 == y + x;

The code:

DeleteDuplicatesBy[listEqs, MemberQ@Level[listEqs, -1]]

picks the first element of the list, which is not what I asked. I want the following list to be the output of the code:

y + z == x^2 + z,
a + b == k + g

Basically, I just want to pick the non-identical equations from the list. The desired equations can be placed in the middle, end or beginning of the list. The position of the equations are not the issue.

Maybe

DeleteDuplicatesBy[listEqs, Reduce] (* or *)
DeleteDuplicatesBy[listEqs, Sort]

x + y == x^2 + z, y + z == x^2 + z

For the longer input list:

DeleteDuplicatesBy[listEqs2, Reduce] 

y + z == x^2 + z, a + b == g + k, x + y == x^2 + z

In both cases, I can't see how x + y == x^2 + z and y + z == x^2 + z can be "equivalent"`.