YES

1 decompositions
 #0 -----------
   1: f(g(x),g(y)) -> f(g(x),h(y))
   2: f(h(x),g(y)) -> f(g(x),g(y))
   3: f(g(x),h(y)) -> f(x,y)
   4: f(h(x),h(y)) -> f(y,x)
   5: f(x,y) -> f(y,x)
   6: g(x) -> h(x)
   7: h(x) -> g(x)

@Rule Labeling
--- R
   1: f(g(x),g(y)) -> f(g(x),h(y))
   2: f(h(x),g(y)) -> f(g(x),g(y))
   3: f(g(x),h(y)) -> f(x,y)
   4: f(h(x),h(y)) -> f(y,x)
   5: f(x,y) -> f(y,x)
   6: g(x) -> h(x)
   7: h(x) -> g(x)

--- S
   1: f(g(x),g(y)) -> f(g(x),h(y))
   2: f(h(x),g(y)) -> f(g(x),g(y))
   3: f(g(x),h(y)) -> f(x,y)
   4: f(h(x),h(y)) -> f(y,x)
   5: f(x,y) -> f(y,x)
   6: g(x) -> h(x)
   7: h(x) -> g(x)