Velocity reciprocity means that the velocity $v$ of a reference frame $R$ with respect to another reference reference frame $R'$ is the opposite of the velocity $v'$ of $R'$ with respect to $R$, i.e. $v'=-v$.
I would like to learn about the experimental evidence - in particular, is it really possible to compare the directions of $v$ and $v'$ or can we only conclude that \begin{equation} \|v\|=\|v'\| \end{equation} (i.e. that the speeds associated to $v$ and $v'$ are equal)?
If we want to be pedantic about it (and if we don't want to say that the question is trivial), the statement of "velocity reciprocity" does not make sense on its face in a fully general setting: Mathematically, positions are points in the space(time) manifold, and velocities are tangent vectors attached to these points; saying that $v_1 = -v_2$ for two velocity vectors only makes sense when they're attached to the same point, otherwise these are simply incomparable objects. When you attach reference frames to two moving different objects, unless the objects are currently colliding, we therefore first need to establish what the statement that their velocities are "equal" actually means:
What allows us to compare tangent vectors at different points is parallel transport. In general, this transport depends on a path, but in flat space, it does not, and this explains why all this stuff about tangent vectors usually doesn't play a role in introductory physics course: In flat space, we can just identify all the tangent spaces with each other, since there's a unique flat parallel transport operator between them.
In curved space, the question becomes ill-defined as there is no unique definition of relative velocities in that case since parallel transport varies with the path we might choose (see e.g. this excellent answer by A.V.S).
So, assuming we're in flat space, there isn't really any problem: The simplest way to implement parallel transport there is just to draw a grid of straight lines (i.e. geodesics) on the floor. Then both your frames have a coordinate system given by the tangent vectors of these geodesics (and additionally the grid makes measuring velocity easy with a stopwatch and some geometry), and you can tell the people in the frames to measure velocity and you'll get $v_1 = -v_2$.
What do you call "experimental evidence" in an experiment you dicide what is the positiv direction. So if you move say in a train with v in positive direction , the trees along the road move from your standpoint in negative direction (or backwards) with the same speed so train R, trees R' than v'=-v
Consider two railway carriages, $A$ and $B$ which were reference frames $A$ and $B$ respectively, moving towards one another on a straight railway track aligned in $\hat x$ direction.
By using a radar type device situated on carriage $A$, the speed of approach of carriage $B$ was measured to be $v_{\rm BA}$ in the $-\hat x$ direction so the velocity of carriage $B$ relative to carriage $A$ is $v_{\rm BA}(-\hat x) = -v_{\rm BA}\hat x$.
While the measurements were going on in carriage $A$ the same set of measurements were also going on with in carriage $B$ with the result that the velocity of carriage $A$ relative to carriage $B$ was found to be $v_{\rm AB}\hat x$.
Those doing the experiment then compared results and found that $v_{\rm BA} = v_{\rm AB}$ which is the experimental evidence that you required.
I think I know now, what your concern is. In the claim v'=-v it is taken for granted that both systems have the same coordinate system, so for both for example on earth the direction going north is positiv, going south is negativ. If R and R' have different definition of positiv direction , one can not compare the sign of velocity. So first the two moving systems have to agree what is the positiv direction, only than the sign of velocity makes sense. And than it is easy to see that the have opposite signs.