Physical interpretation of 4th-order correlations
If you are willing to relax the WSS assumption, and consider cyclostationary signals, you might check out my theory of pure and impure sine-wave components of higher-order probabilistic parameters:

On pure and impure sine-waves, higher-order moments, and cyclic cumulants

and

Oddities concerning pure nth-order sine waves

I've thought about your question a lot, more so in the past than lately, and all I've been able to come up with is that higher-order moments contain characteristic sine-wave components (characteristic of the process, the order, and the number of conjugated factors), and higher-order cumulants characterize what is new in the moment that cannot be accounted for by products of lower-order moments. Unlike the easy physical interpretation of autocorrelation as power (think of the zero-lag autocorrelation), I don't have a simple physical unit to assign here, but maybe my hint will lead you to a more satisfying answer.