Measures of Center/Spread in Categorical/Ordinal

[WWGD]

Hi All,
I am kind of confused about whether mean and sd can/are used as measures of center, spread respectively when dealing with Ordinal variables.
On one hand, this choice does not seem to reasonably support the interpretation. It seems that median and mode .OTOH, I have seen it used in some papers, although I am not sure how reputable are the journals where these papers were published. EDIT: Are Mode and Median the standard measures of center and spread for Ordinal Categorical variables?

 

[StoneTemplePython]

Are we just talking $\{0,1\}$ or perhaps $\{-1,1\}$ or whatever binary encoding, or something more? It seemed odd when I first saw it, but I quite like $\{-1,1\}$ -- if the mix of positives and negatives is even, you get something with zero mean, which is nice.

In any case, if it is just bools, then there's one sort of inference we can do.

If its more than 2 possible values, how can median and mode give you measures of spread?

Other thoughts:

1.) are we talking about variables / features, or something different like labels (i.e. the thing you want to estimate in supervised learning)? I am hoping we're talking features / dimensions to the data not labels.

2.) Btw, I haven't done all that much in this regard, but one nice thing to talk about is entropy. All you need is probabilities / percentage mix for each of those ordinal values for a given feature-- and you get a pretty nice way of discussing spread / dispersion.

3.) I generally like CDFs and the median sounds fine in this regard. I'm not so sure about it being the 'center' though.

4.) sometimes reducing things to the boolean case is useful ('one hot encoding' is a term that certain people like to use). It's basically an indicator variable expansion. Once you have it in a suitable boolean setup, I think it's not uncommon to act as if the ranking is cardinal as opposed to ordinal... and again throw in a continuity relaxation and run regression or whatever on it.