I have seen in a variety of texts that an $n$-differentiable function $f$ is written \begin{align} f\in C^{n}\Longleftrightarrow f^{\left(n\right)}\in C,\tag{1} \end{align} such as in Widder's Advanced Calculus. What if I wished to work with an "$n$-integrable" function? Would I introduce it as \begin{align} f\in C^{-n}\Longleftrightarrow \overbrace{\idotsint}^{n}f\in C?\tag{2} \end{align} Or perhaps some variant of that? I haven't taken Real Analysis yet, so forgive me if it's completely obvious. I just haven't seen authors mention it in any text I've seen (probably because this just isn't the way to write it).
EDIT: Widder introduces $C$ as the class of continuous functions.
Integration is, in a sense, "less sensitive" that differentiation. Whereas when we take derivatives we expect continuity, we actually do not require continuity to compute integrals.
The term "integrable" is typically used to denote functions whose integral over some domain is finite, e.g. $\int_{-\infty}^\infty f(x)\, dx < \infty$ means that $f$ is integrable (implicitly, with respect to Lebesgue measure).
But even then, we tend to restrict the term to mean functions whose absolute values are integrable, i.e. $\int_X |f| < \infty$ on a measure space $(X,\mathcal{M},\mu)$. This is a stronger statement than the previous: if $\int_X |f| < \infty$, then we must have $\int_X f^+ < \infty$ and $\int_X f^- < \infty$, where $f^+$ and $f^-$ are the positive/negative parts of $f$.
We use a notation for such functions: $f \in L^1$, or sometimes $f \in L^1(X,\mu)$, if the space and measure need explicitly be satisfied.
Measure theoretic (i.e. Lebesgue integration) is somewhat stronger than the integration you might be familiar with, which is called "Riemann integration." Except in pathological cases, Lebesgue integrability implies Riemann integrability. The interesting thing about Lebesgue integrability is that we can allow the integrand to have infinitely many discontinuities, pursuant to certain conditions.
What this means is that we can have a function that is Riemann integrable but it is ill-behaved in a topological sense. In fact, if we took $F(x) = \int _a^x f(t)\, dt$, then it is not necessarily true that $F'(x) = f(x) - f(a)$ everywhere -- the actual term used is almost everywhere.
Now, this said, you asked about computing something like $\int \int f(x) \, dx^2$. If this integral exists, and if $f \in L^1$, we don't say that $f \in L^2$, at least not for this reason.
There is a theorem that deals with this condition. The theorem is known as the Fubini-Tonelli theorem and it explores when we can look at an integral over a product measure as an iterated integral. Any such function where $\int \int f\, dx^2$ exists and was finite would require that $f$ is in $L^1$, or that $f$ is a non-negative measurable function (as well as that our measure space is $\sigma$-finite). However, it is quite possible to consider functions that are $L^1$ integrable and where $\int \int f\, dx^2$ exists, but $f \notin L^2$.
In short, we don't look at integration as "backwards differentiation" per se. While we do talk about spaces of integrable functions, $L^1, L^2, L^p,$ etc, this term is not akin to "differentiable" in the sense that we use in calculus.
Now, this is only intended to be a very broad overview of integration, and there are many, many, many subtleties to all of this. I did not intend to capture every detail. In fact, we can even talk about measure-theoretic derivatives that don't have the pointwise sensitivity that traditional derivatives have. But we handle them a bit differently, and typically $C^k$ means the space of $k$-times continuously differentiable functions in the conventional sense.