Turbulent combustion involves many interdependent variables evolving over a wide range of length and time scales. Due to the tremendous computational cost required to solve equations for all of these variables over all relevant length scales, reduced-order models are required for predictive simulations of engineering systems. Common approaches to reduced-order combustion modeling include physically-informed methods of dimension reduction, such as Flamelet Generated Manifolds (FGM) and the Flamelet/Progress Variable model (FPV), and, more recently, data-driven dimension reduction, such as Principal Component Analysis (PCA). While the accuracy of both approaches has been previously assessed, they have not been directly compared or used in conjunction with one another. To this end, this work examines both how insights from physically-derived reduced-order manifolds (PDROM) can be used to benchmark data-driven approaches and how the data-driven approaches can be used to validate the assumptions of PDROM models, using data from Direct Numerical Simulations (DNS) of premixed and nonpremixed turbulent flames and from one-dimensional flame calculations. PCA assumes a linear structure and therefore cannot achieve as large of a dimension reduction as PDROM while maintaining accuracy. While nonlinear dimension-reduction methods perform similarly to PDROM, these methods suffer from a lack of physical interpretability that would make implementation impractical. However, the data-driven approaches are successfully used to assess the critical assumptions of the physical models, with the similarity of principal components between different data sets validating structural similarity between the DNS and one-dimensional flame data.