diff --git a/src/comparison.jl b/src/comparison.jl
index 258749cc..9a3ce0e4 100644
--- a/src/comparison.jl
+++ b/src/comparison.jl
@@ -307,14 +307,15 @@ ordering(p::AbstractPolynomialLike) = ordering(typeof(p))
 # of x < y is equal to the result of Monomial(x) < Monomial(y)
 # Without `Base.@pure`, TypedPolynomials allocates on Julia v1.6
 # with `promote(x * y, x)`
-Base.@pure function Base.cmp(
-    ::AbstractMonomialOrdering,
-    v1::AbstractVariable,
-    v2::AbstractVariable,
-)
+Base.@pure function Base.cmp(v1::AbstractVariable, v2::AbstractVariable)
     return -cmp(name(v1), name(v2))
 end
 
+function Base.cmp(m1::AbstractMonomial, m2::AbstractMonomial)
+    s1, s2 = promote_variables(m1, m2)
+    return cmp(ordering(m1)(), exponents(s1), exponents(s2))
+end
+
 function compare(
     m1::AbstractMonomial,
     m2::AbstractMonomial,
@@ -335,22 +336,14 @@ end
 # less than `b`, they are considered sort of equal.
 _cmp_coefficient(a, b) = 0
 
-function Base.cmp(
-    ordering::O,
-    t1::AbstractTermLike,
-    t2::AbstractTermLike,
-) where {O<:AbstractMonomialOrdering}
-    Δ = cmp(ordering, monomial(t1), monomial(t2))
+function Base.cmp(t1::AbstractTermLike, t2::AbstractTermLike)
+    Δ = cmp(monomial(t1), monomial(t2))
     if iszero(Δ)
         return _cmp_coefficient(coefficient(t1), coefficient(t2))
     end
     return Δ
 end
 
-function Base.cmp(t1::AbstractTermLike, t2::AbstractTermLike)
-    return cmp(ordering(t1)(), t1, t2)
-end
-
 Base.isless(t1::AbstractTermLike, t2::AbstractTermLike) = compare(t1, t2) < 0
 
 _last_lex_index(n, ::Type{LexOrder}) = n
diff --git a/test/commutative/comparison.jl b/test/commutative/comparison.jl
index 94a5baf9..b7e12313 100644
--- a/test/commutative/comparison.jl
+++ b/test/commutative/comparison.jl
@@ -141,28 +141,22 @@
         @test p !== nothing
         @test p !== Dict{Int,Int}()
     end
-    @testset "compare" begin
-        lex = LexOrder
-        grlex = Graded{lex}
-        rinvlex = Reverse{InverseLexOrder}
-        grevlex = Graded{rinvlex}
-        Mod.@polyvar x y z
-        # [CLO13, p. 58]
-        @test compare(x * y^2 * z^3, x^3 * y^2, lex) < 0
-        @test compare(x * y^2 * z^3, x^3 * y^2, grlex) > 0
-        @test compare(x * y^2 * z^3, x^3 * y^2, rinvlex) < 0
-        @test compare(x * y^2 * z^3, x^3 * y^2, grevlex) > 0
-        @test compare(x * y^2 * z^4, x * y * z^5, lex) > 0
-        @test compare(x * y^2 * z^4, x * y * z^5, grlex) > 0
-        @test compare(x * y^2 * z^4, x * y * z^5, rinvlex) > 0
-        @test compare(x * y^2 * z^4, x * y * z^5, grevlex) > 0
-        # [CLO13, p. 59]
-        @test compare(x^5 * y * z, x^4 * y * z^2, lex) > 0
-        @test compare(x^5 * y * z, x^4 * y * z^2, grlex) > 0
-        @test compare(x^5 * y * z, x^4 * y * z^2, rinvlex) > 0
-        @test compare(x^5 * y * z, x^4 * y * z^2, grevlex) > 0
-        # [CLO13] Cox, D., Little, J., & OShea, D.
-        # *Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra*.
-        # Springer Science & Business Media, **2013**.
+    lex = LexOrder
+    grlex = Graded{lex}
+    rinvlex = Reverse{InverseLexOrder}
+    grevlex = Graded{rinvlex}
+    @static if Symbol(Mod) == :DynamicPolynomials
+        @testset "compare $M" for M in [lex, grlex, rinvlex, grevlex]
+            Mod.@polyvar x y z monomial_order = M
+            # [CLO13, p. 58]
+            sgn = (M == lex || M == rinvlex) ? -1 : 1
+            @test sgn * compare(x * y^2 * z^3, x^3 * y^2) > 0
+            @test compare(x * y^2 * z^4, x * y * z^5) > 0
+            # [CLO13, p. 59]
+            @test compare(x^5 * y * z, x^4 * y * z^2) > 0
+            # [CLO13] Cox, D., Little, J., & OShea, D.
+            # *Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra*.
+            # Springer Science & Business Media, **2013**.
+        end
     end
 end