Recently during a lecture, my professor mentioned that forcing over any poset which is countable, separative, and atomless, is essentially the same as forcing over the Cohen poset, that is to say results in adding a Cohen real.

My question is: Are there any other similar characterizations of "commonly used" forcing posets? Specifically, is there one for the Hechler poset?

The Hechler Poset/forcing notion $(H,\le)$ is given by setting $H=\omega^{\lt\omega} \times \omega^{\omega} $, and defining the relation $(t,v) \le (s,u)$ iff $ ( t \supset s \wedge (\forall n\in\omega) (u(n) \le v(n)) \wedge (\forall m \in dom(t \backslash s))(t(m) \gt u(m))$. When forcing with this poset, you end up adding an unbounded real to the ground model.

I understand that you cannot produce a model in which $\mathfrak{b}=\omega_2$ using product forcing, and that you need iterated forcing to do so. Moreover, the iterated forcing construction I've seen that produces $\mathfrak{b}=\omega_2$ in the forcing extension, used the finite support iteration of $\omega_2$ many copies of the Hechler poset. Is this evidence for the lack of such a characterization?

(I apologize in advance if this is an ill-stated question, I will change it accordingly if it is.)