# Difference-in-differences with no pre-treatment?

The typical difference-in-differences estimator (as fixed effects) fits a model of the form $$y_{it} = \alpha_i + \delta T_{it} + X_{it}'\beta + \epsilon_{it}$$

where $T$ is some treatment that happens to $i$ at time $t$.

The coefficient $\delta$ is identified from the jump between time periods when T goes from zero to one, essentially using as counterfactuals those that didn't get treated during that period, after controlling for unobservables that don't vary in time.

Normally the (panel) dataset starts with everyone un-treated, and ends with some remaining untreated while others get treated. Alternatively, if everyone (eventually) gets treated, you can still include post-treatment data to improve statistical precision -- the $\delta$ is still identified from the time periods where some got treated and others didn't.

My question: is it legit to fit a model where one group starts treated, the other group starts untreated, and then the untreated group gets treated? This is basically the mirror image of a situation in which one group stayed untreated and one group got treated -- we still have heterogeneity in some time periods. Mathematically it seems identical -- the standard error components motivations seems to still apply.

Am I missing something?