Suppose the following integral $$ \int\limits_{-\infty}^{\infty}\sin{x}dx $$ In mathematical rigor, the following is the definition $$ \int\limits_{-\infty}^{\infty}\sin{x}dx = \lim\limits_{n\to\infty}\lim\limits_{m\to-\infty}\int\limits_m^n\sin{x}dx $$
Some places (especially in physics), say that since $\sin{x}$ is an odd function, then the integral is $0$ since the interval is symmetric.
That sounds pretty weird to me. The above limit is like the limit of the the series $2,0,2,0,2,0,\ldots$(if jumping in intervals of $\pi$) which obviously diverges. How can I know that $m,n$ tend to $\pm$ infinity in a symmetric way? Is this the general case?
You are correct, and others are wrong to say it converges to zero. As pointed out by 5xum, there are some people who want to integrate only over symmetric intervals, but they should not use the same notation for such an "integral", precisely due to the fact that your objection is valid and on point. Perhaps a notation and definition somewhat like the following would be better (or is already somewhere in use):
$$ \rlap{\int_a^b}\hspace{2pt}{\large \sigma}\quad f(x)\ dx\triangleq\frac12\left(\lim_{\alpha\to a}\left[\int_{\alpha}^{\frac{\alpha+b}2}f(x)dx+\int_{\frac{\alpha+b}2}^bf(x)dx\right]+\lim_{\beta\to b}\left[\int_a^{\frac{a+\beta}2}f(x)dx+\int_{\frac{a+\beta}2}^{\beta}f(x)dx\right]\right). $$
Here, I have used the greek letter $\sigma$, superimposed on the integral sign, to mean a "symmetric" integral, in a sense close to the one mentioned (so that in that special case, my suggestion reduces to that one), but for integrals over any real interval for which the integrals over all of the symmetric subintervals of the kind indicated converge to something. Maybe this can fail to capture the desired variant of an integral if the physicists mentioned by 5xum must be over intervals of the form $[-n,n]$ where $n$ is a positive integer. I've not checked to try to find an example for which $\displaystyle\lim_{n\to\infty}\int_{-n}^n\ f(x)\ dx=0$ but $\displaystyle \rlap{\int_{-\infty}^{\infty}}\hspace{2pt}{\large \sigma}\quad f(x)\ dx$, as defined above, is a divergent improper "symmetric integral". Such a monster would be interesting, I'd think, and may be already known to someone other than me. I also have not carefully checked that the kinds of symmetric integrals you asked about are a special case of what I defined above. I can look into that later, perhaps, or someone else can, if they wish.
I don't know what Cesaro integration (mentioned by Anixx) is yet, but I'd like to know (I came to this page while searching for a definition of cesaro integration). I cannot yet comment on other answers because I have yet to garner enough rep to do so... More answering needed I guess...
A "probabilistic" (actually, measure-theoretic) variant of the above might be interesting as well:
For nonnegative functions $f$ such that the integrals converge, and for finite $a$ and $b$, $$ \rlap{\int_a^b}\hspace{0.5pt}\stackrel{\large\sigma}{\mu}\quad f(x)\ dx\triangleq\lim_{\alpha\to 0}\sup\left\{\left.\int_{E}f(x)dx+\int_{F}f(x)dx\right\vert E\subseteq (a,b), \mu[(a,b)\setminus(E\cup F)] + \vert\mu(E)-\mu(F)\vert<\alpha\right\}, $$ and for other functions $f=f^++f^-$ where $f^+=\sup(f,0)$ and $f^-=\inf(f,0)$ are the nonnegative and nonpositive parts of $f$, respectively, $$ \rlap{\int_a^b}\hspace{0.5pt}\stackrel{\large\sigma}{\mu}\quad f(x)\ dx\triangleq\rlap{\int_a^b}\hspace{0.5pt}\stackrel{\large\sigma}{\mu}\quad f^+(x)\ dx-\rlap{\int_a^b}\hspace{0.5pt}\stackrel{\large\sigma}{\mu}\quad -f^-(x)\ dx, $$ or some such. There should be many ways to define integrals that have a "symmetric integration" kind of flavor, with various kinds of uses in different settings. Someone probably has taken these approaches I've mentioned before and either decided they don't help with their situation. or that they do, and I don't know who these people would be, but in part, I'm suggesting that anyone who does so needs to "decorate" their integral signs in a way that helps avoid confusion and contradictions due to conflation of concepts or lack of rigor.