We present the bilinear phase map, a concept that generalizes the Kramers-Wannier transformation to investigate unconventional gapped phases of qudit spin chains. Encoding the transformation in a matrix, the bilinear phase map enables the exploration of a broader spectrum of generalized quantum phases and dualities, including the nonunitarity in duality transformations, and derivation of general noninvertible fusion rules. Furthermore, we obtain strong constraints on anomaly conditions of qudit models with a general class of duality symmetries. This establishes a microscopic anomaly classification for a broad class of lattice dualities.

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• Quantum criticality
• Symmetry protected topological states
• Topological order
• Topological phase transition
• Duality
• Gauge symmetries
• Quantum spin chains
• Spin chains