# How does the symmetry of indices of tensors work?

Very quick and short clarification:

Knowing $$R_{abcd} = - R_{bacd}$$, is this symmetry true for for a,b components (i.e $$R_{acbd}=-R_{bcad}$$) or for the first two 'slots' of the tensor (i.e $$R_{acbd}=-R_{cabd}$$). This is specifically for the Riemann Tensor if that matters, although I assume there is a strict convention.

I am sure this is a very basic fact about how tensors transform but I haven't found a clear "this is how it works".

The letters are just there to keep track of each entry, they generically do not represent anything by themselves (unless you are prescribing them a specific value, such as $$a = r$$, where $$r$$ is the radial coordinate). As a consequence, the symmetry applies to the slots, not to the entries.
In my opinion, Geroch's notes on Differential Geometry (available for free on his website) are particularly good to get a grip with this notation. In Chap. 4 he discusses how abstract index notation works in detail. In short, the indices play a role similar to that of the little arrow when we write $$\vec{v}$$, but in here they also keep track of slots.