Is the notation $\int_{\Bbb R}$ equivalent to $\int_{-\infty}^\infty$? The question is pretty straightforward. In class, I have seen $\int_{\Bbb R}$ and $\int_{-\infty}^\infty$ being used interchangeably. However $\int_{\Bbb R}$ contains no implicit sense of direction, so technically isn't $\int_\infty^{-\infty}$ also the same as $\int_{\Bbb R}$? 
Or $\int_{\infty}^{0}+\int_{-\infty}^{0}\equiv\int_{\Bbb R}$, provided all integrals exist? 
There could be several such possibilities. Is that why we have attached a conventional direction to $\int_{\Bbb R}$, i.e. from $-\infty$ to $\infty$?

The notation $\int_a^b$ (or, in an extended sense, $\int_{-\infty}^{\infty}$) is meant to be suggestive of Riemann integration, which, as you said, entails an orientation in your Riemann sums and hence, your integrals.
The notation $\int_{\mathbb{R}}$ is a more general notation from the theory of Lesbegue integration, where integrals are non oriented and over a measure space $X$. You are, indeed, correct that $\int_{-\infty}^{\infty}$ and $\int_{\mathbb{R}}$ are used interchangably, but this is due to the fact that the Lesbegue integrals and the Riemann integrals agree in pretty much all relevant cases. The notation $\int_{\infty}^{-\infty}$ refers, conventionally, to "summing in the wrong direction" and it's not true that this could be substituted with $\int_{\mathbb{R}},$ since the corresponding Riemann and Lesbegue integrals will have a sign difference.
