First and second order statistics

I know that this is an old question with an accepted answer. But in the hope of helping someone finding this question now or in the future I'm answering.

I believe the paper was referring to "low-order statistics" and not "order statistics". Low-order statistics are functions or quantitative measures which use the zeroth, first and second power of a sample.

For a probability density, the first-order statistic (first moment) is the mean, the second-order statistic (second moment) is the variance.

On the surface, it seems like the point they're trying to make has more to do with identifiability than order statistics. Perhaps, in their language, "order statistics" refers to moments (in statistics) or powers of a sample. This is evidenced by their statement in parentheses about the mean and covariance.

If they are actually referring to moments, and a finite set of moments drive the states in question (e.g. as a normal distribution is fully determined by its mean and [co]variance), then they might simply be describing the model's inherent symmetry and/or limitations by way of its low-order moments.

Also, it sounds like they're voicing concerns similar to those that arise when considering the identifiability of mixture models.