  We extend some classical inequalities between the Dirichlet and Neumann
eigenvalues of the Laplacian to the context of mixed Steklov--Dirichlet and
Steklov--Neumann eigenvalue problems. The latter one is also known as the
sloshing problem, and has been actively studied for more than a century due to
its importance in hydrodynamics. The main results of the paper are applied to
obtain certain geometric information about nodal sets of sloshing
eigenfunctions. The key ideas of the proofs include domain monotonicity for
eigenvalues of mixed Steklov problems, as well as an adaptation of Filonov's
method developed originally to compare the Dirichlet and Neumann eigenvalues.
