  In this paper we investigate the (classical) weights of mod $p$ Siegel
modular forms of degree 2 toward studying Serre's conjecture for $GSp_4$. We
first construct various theta operators on the space of such forms a la Katz
and define the theta cycles for the specific theta operators. Secondly we study
the partial Hasse invariants on each Ekedahl-Oort strata and their local
behaviors. This enable us to obtain a kind of weight reduction theorem for mod
$p$ Siegel modular forms of degree 2 without increasing level.
