Biological cycles such as the cell cycle and circadian rhythms are fundamental cell-autonomous processes that are often obscured in single-cell RNA sequencing (scRNA-seq) data by other sources of variation, including cell-type identity, metabolic or disease-associated states. Disentangling these continuous periodic trajectories from other forms of cellular variability remains a major challenge in single-cell analysis. Here, we introduce CoPhaser, an algorithm designed to learn context-dependent periodic manifolds that decompose scRNA-seq count data into independent periodic and non-periodic sources of variation, while maintaining interpretability of manifold coordinates across diverse biological contexts. CoPhaser employs a biologically informed variational autoencoder with a structured latent space that explicitly separates cycle phases from cellular context, while controlling their mutual information. By modeling gene expression as context-modulated harmonic functions, the model captures flexible but biologically motivated deformations of periodic manifolds. We demonstrate that CoPhaser recovers biologically accurate continuous cell-cycle phases across diverse RNA sequencing technologies, including highly heterogeneous situations such as development and cancer, without requiring prior knowledge of gene expression programs or cell-cycle states. In two cancer applications, CoPhaser reveals subtype-specific proliferation dynamics, identifying quiescent primitive states in relapsed pediatric acute myeloid leukemia and distinguishes proliferation-driven gene overexpression from constitutive overexpression to uncover potential robust therapeutic targets in triple-negative breast cancer. CoPhaser readily generalizes to other periodic systems, enabling the reconstruction of circadian clocks in the mouse aorta and the identification of cell-type and subtype-specific circadian differences. Furthermore, it maps continuous endometrial remodeling across the human menstrual cycle, uncovering distinct transcriptional dynamics between women with endometriosis and controls. Together, CoPhaser provides a versatile and interpretable framework for dissecting the interplay between cellular identity and biological cycles in single-cell data.