How to interpret a polychoric correlation? I've checked related links to understand what a Polychoric Correlation measures, got some 404's and other unanswered questions. I've seen it used to determine independence between Nominal/Interval ( "Contunuized"*, aka, Jointly Normal ) on YouTube (https://www.youtube.com/watch?v=besaBez9giw&t=597s), as ways of measure correlation between (underlying continuous ) variables, and I've seen it used to describe the distribution of some (underlying continuous ) trait of fertility of cattle given data on cattle births (From John Uebersax' page ).
I (think I) understand the general underlying setup for the polychoric. We start with a paired
discrete/Interval/Ordinal data set . We assume each data set is discrete output comes from an
underlying Normal Distribution , and the pair of variables is jointly-normal. The poly/tetra choric
is then the Pearson correlation between these underlying continuous variables . Is this correct? If so, does it have / can it be used for the purposes described in the above paragraph?
Edit: Also, are normality of each and joint normality required? How robust is it (i.e., how well does the test fare under departures from normality assumptions). Because doesn't joint normality equivalent to individual Normality?

The answer by Gregg H is very good. However I'd like to add something about the "assumption" of normality. The whole concept of polychoric correlation is based on the normality assumption. In fact the same observable distribution of categories is compatible with any continuous marginal distribution, and I'm pretty convinced that often the correlation between two latent variables can take a large range of values if we try to find "worst case" non-normal underlying distributions that move the correlation as far away from the one under normality assumption as possible. This is because the correlation is very sensitive to outliers, so that shifting small probabilities to infinity in the latent distribution will have a huge impact on correlations. (Note that for sure the latent distributions can be chosen so that second moments, and therefore correlations, do not exist.)
So statements about the "true" correlation between the "true" latent variables will not be robust, and one can't just justify using the normality assumption for convenience reasons. There is no way to test it either, as Gregg H correctly noted. In fact it is possible to test whether the dependence structure is like a normal one (Gaussian copula), but this is not enough to make sure that correlations are robust, as outliers can spoil the correlation through the marginal distributions even if the dependence structure is fixed.
This seems quite negative, however my own interpretation would be different. I do not think that it is in fact required that the "true" underlying distribution is Gaussian; I'd rather say that the polychoric distribution is a measure of correlation between the observed categorical variables that is "calibrated" through an underlying Gaussian model. The role of the Gaussian model here is not to be "true", but rather to fix a setup that allows to compute a correlation measure. The Gaussian is symmetric and does not have heavy tails, so it will lead to a rather regular and intuitive correlation measure; skew distributions or distributions with outliers will imply correlations that are less intuitively connected to the observed categorical dependence, therefore less desirable. (In fact I believe that the Gaussian correlation will be more robust, at least regarding what happens working with similarly "nice" distributions, than any correlation based on a skew or heavy-tailed distribution, even though worst cases will always be bad).
So there is a way to use polychoric correlations that in my view makes sense regardless of whether there is a "true latent distribution". In most cases I'd say that such a distribution does not really exist, rather it is a modelling device, and then we can choose a nice and convenient one. However, beware of interpreting polychoric correlations as true correlations between real latent variables.
