We establish scaling limit theorems for the up-down ordered Chinese restaurant processes (oCRPs) of Rogers and Winkel as processes in a space of interval partitions. As previously conjectured, the limits are self-similar diffusions previously constructed directly in the continuum. We extend the oCRP model and the results to a three-parameter family ${\rm oCRP}^{(α)}(θ_1,θ_2)$, $α\in(0,1)$, $θ_1,θ_2\ge 0$. We use the scaling limit approach to extend existing stationarity results to the full three-parameter family, identifying an extended family of Poisson--Dirichlet interval partitions. Their ranked sequence of interval lengths has Poisson--Dirichlet distribution with parameters $α\in(0,1)$ and $θ:=θ_1+θ_2-α\ge-α$, including for the first time the usual range of $θ>-α$ rather than being restricted to $θ\ge 0$. This has applications to Fleming--Viot processes, nested interval partition evolutions and tree-valued Markov processes, notably relying on the extended parameter range.