Cross-subject EEG decoding remains a fundamental challenge due to substantial inter-subject variability in brain activity, which hinders the development of subject-independent EEG models. Despite progress in extracting cross-subject invariant features, existing studies neglect the shared neural responses that arise under similar cognitive or emotional states across individuals, limiting their ability to learn generalized and consistent EEG representations. To address the challenges, we propose State Mamba, a novel spatiotemporal EEG state-space model that explicitly models and aligns neural responses and their spatiotemporal state transitions to learn consistent and generalizable representations across subjects. Innovatively, State Mamba theoretically formulates a multi-channel Mamba architecture that jointly models spatial and temporal brain state transitions, supporting principled analysis of neural responses. To enhance spatiotemporal feature coupling, we introduce the LGANN module, which adopts global-local attention to integrate long- and short-term brain activity into a compact EEG representation. Furthermore, we design two self-supervised pretext tasks to extract consistent neural patterns across subjects: (1) representation alignment to align EEG representation, and (2) pattern alignment to align their transition rules under identical conditions, jointly promoting subject-invariant EEG representations. Extensive experiments on three benchmark datasets, FACED, DEAP, and ISRUC, demonstrate the superior performance of State Mamba in cross-subject emotion and sleep recognition tasks, validating its robust generalization capability.