How to Deal with an Ordinal Dep Var When You Need/Want to Include Fixed Effects?

A recent question posed on StataList has stimulated an interesting discussion (here) as well as a call for FE ordinal models. The specific question:

"So to sum up my question: Is it possible to 'argue' that my ordinal dependent variables can be treated as continuous and I can therefore use FE for them as well? If yes - could you point me towards the appropriate literature/methodology? Or is this not possible and I need to use logit/probit for the ordinal dependent variables?"

The core of the question--can an ordinal dependent variable be treated as "continuous"--comes up most frequently in the survey context where a response involves a Likert Scale (typically a 5- or 7-point scale -- or an even-numbered scale that pushes respondents off a mid-point response). While reasonable minds can--and do--differ on what modeling options may plausibly exist, at the very least, and as a first step, a researcher would need to closely examine how any ordinal dependent variable distributes as this may itself limit model selection options.