Another take on Omar Nagib's Answer is that one justifies the use of velocity $-v$ for A relative to B whenever we use $v$ for B relative to A by what is sometimes (rather vaguely and unhelpfully) the reciprocity relationship.
As Omar notes this relationship is not particular to special relativity but also applies to the Galilean velocity addition (infinite $c$) relativity. And the fundamental reason is this:
Galileo's principle (of indetectability of uniform motion of a frame from within a frame) implies the transformations between inertial frames together with transformation composition form a group. Copernicus's principle that spacetime is homogeneous implies that the transformations act linearly on affine co-ordinates. So we have a matrix group action whereby
where $K$ and $\exp(\eta\,K)$ are $2\times 2$ matrices and $\eta$ encodes the relative motion's "swiftness" (think of it as a generalized velocity - an invertible function of the time over distance velocity).
Now, by spatial isotropy, we can now choose the $-x$ direction to be our positive spatial direction. There's nothing fundamentally different about this direction compared with the $+x$ direction, so if we reason above, we must get the same $K$ matrix as above, and a possibly different $\eta$ parameter, say $\eta^\prime$.
Thus, when we make the transformation $x\mapsto-x$, we must have:
for all $x$ and $t$. A little bit of algebra then shows that $\eta^\prime= -\eta$ and, on "calibrating" the $\exp(\eta\,K)$ in terms of everyday velocities, we find $v^\prime = -v$.
You might like to read some more details on this in my article "Of Groups, Galileo and what's so special about the speed of light": I've put a preprint on my website and hope it will be published in the European J. Physics sometime soon. "Reciprocity" is also discussed in:
Stefano Liberati, Sebastiano Sonego, Matt Visser, "Faster-than-c signals, special relativity, and causality", Ann. Phys. 298 (2002) pp167-185