*Below, all sentences/formulas are first-order and in the language of arithmetic. For simplicity, we conflate numbers and numerals, and conflate sentences/formulas and their Godel numbers.*

Given a formula $\varphi(x)$ and a sentence $\theta$, say that $\theta$ **asserts its own $\varphi$-ness** iff $\mathsf{PA}\vdash\theta\leftrightarrow\varphi(\theta).$ Let $\mathsf{SR}(\theta)$ be the set of $\varphi$ such that $\theta$ asserts its own $\varphi$-ness. Bounded truth predicates show that $\mathsf{SR}(\theta)$ is never empty. I'm curious how much $\mathsf{SR}(\theta)$ could actually depend on $\theta$; in particular, if there's a lot of potential variety here, this might give a meaningful notion of "degree of self-referentiality" of a sentence.

Here's one way to make this precise:

Are there sentences $\theta,\theta'$ such that $\mathsf{SR}(\theta)\not\cong\mathsf{SR}(\theta')$ as partial orders?

The partial ordering I have in mind is provability: $\sigma\le\rho$ iff $\mathsf{PA}\vdash\forall x[\sigma(x)\rightarrow\rho(x)]$. By considering bounded truth predicates, $\mathsf{SR}(\theta)$ always contains a copy of $\mathbb{Z}$: briefly, look at $\tau_n^+(x)=$ "$x$ is a true $\Sigma_n$ sentence" and $\tau_n^-(x)=$ "$x$ is not a false $\Sigma_n$-sentence" for $n$ sufficiently large. Another "canonical" element is the formula $\varphi(x)\equiv\theta$ (basically, "ignore input, output $\theta$"). Finally, $\mathsf{SR}(\theta)$ is always a distributive lattice. Beyond this, however, I don't see anything useful.