We present a detailed molecular dynamics study of water reorientation and hydrogen-bond dynamics in a strong confinement situation, within the narrow pores of an all-silica Linde type A (LTA) zeolite. Two water loadings of the zeolite are compared with the bulk case. Water dynamics is found to be retarded in this extreme hydrophobic confinement and the slowdown is more pronounced at higher water loading. We show that water reorientation proceeds mainly by large-amplitude angular jumps, whose mechanism is similar to that determined in the bulk. The slowdown upon hydrophobic confinement is found to arise predominantly from an excluded-volume effect affecting the large fraction of water molecules lying at the interface with the zeolite matrix, with an additional minor contribution coming from a structuring effect induced by the confinement.