Can There Be Infinitely Many Models Equivalent to a Given Covariance Structure Model?

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Investigating population heterogeneity with factor mixture models. (2005, methodology): “In the context of models for data from single homogeneous populations, the problem of equivalent models has led to a considerable body of research (see, for instance, Hershberger, 1994; Luijben, 1993; MacCallum, Wegener, Uchino, & Fabrigar, 1993; Raykov & Marcoulides, 2001).”
Assessing Mediation in Dyadic Data Using the Actor-Partner Interdependence Model (2011, methodology): “For the unconstrained APIMeM, there are several equivalent models that might suggest very different conclusions ( Bentler & Satorra, 2010 ; Raykov & Marcoulides, 2001 , 2007 ).”
Exploring the factors of gastronomy blogs influencing readers' intention to taste. (2011, background): “According to Raykov and Marcoulides (2001), two models (say model 1 and model 2) can be considered effectively equivalent when model 1 includes a correlation/covariance between two latent variables and in model 2 such a parameter is replaced by a direct path between the same variables.”
Lack of Theory Building and Testing Impedes Progress in The Factor and Network Literature (2020, background): “Second, temporal data can help eliminate some equivalent models, since it makes little sense to orient arrows backwards in time, but such data do not eliminate the problem of statistical equivalence in principle (Raykov & Marcoulides, 2001).”