We prove that the
(
k
+
d
)
(k+d)
-th Neumann eigenvalue of the biharmonic operator on a bounded connected
d
d
-dimensional
(
d
≥
2
)
(d\ge 2)
Lipschitz domain is not larger than its
k
k
-th Dirichlet eigenvalue for all
k
∈
N
k\in \mathbb {N}
. For a special class of domains with symmetries we obtain a stronger inequality. Namely, for this class of domains, we prove that the
(
k
+
d
+
1
)
(k+d+1)
-th Neumann eigenvalue of the biharmonic operator does not exceed its
k
k
-th Dirichlet eigenvalue for all
k
∈
N
k\in \mathbb {N}
. In particular, in two dimensions, this special class consists of domains having an axis of symmetry.