Predicting stress fIelds of porous metamaterials under arbitrary rotations is essential for reliable online monitoring in additive manufacturing. However, existing neural operators are limited by two major issues: geometric orientation bias where identical lattices are misinterpreted under rotation, and non-physical stress leakage caused by the failure of Partial Differential Equation (PDE) constraints at discontinuous void-solid interfaces in porous metamaterials. To address these challenges, we propose the Symmetry-Imbued Hamiltonian Neural Operator (SIHNO) that embeds geometric symmetry and an energy-structured inductive bias into a unified neural operator architecture. Specifically, SIHNO introduces a Rotation-Steerable Lattice Projector (RSLP) that lifts lattice images into a continuous-angle geometric symmetry representation. This geometric representation is further coupled with a Hamiltonian-Fueled Propagator (HFP) that replaces local PDE constraints with Hamiltonian-inspired energy propagation. Finally, a slice-aware convolutional decoder reconstructs stress fields based on RSLP and HFP. Comprehensive experiments demonstrate that SIHNO outperforms existing neural operators. The proposed framework provides a robust architecture for stress-field prediction in additive manufacturing.