Consider the group $G := \langle (1,3), (1,2,3,4) \rangle$. This group is of order $8$ and its elements are
$$G := \{ (), (1234), (24), (12)(34), (13), (14)(23), (13)(24), (1432) \}.$$
Each of these elements have the property that whenever it is applied to each integer in $\{1,2,3,4\}$, the 'gap' between $1$ and $3$ (and similarly $2$ and $4$) is always a multiple of $2$. For example starting off with $[1,2,3,4]$ and applying $(1234)$ gives $[2,3,4,1]$, and then applying $(14)(23)$ gives $[3,2,1,4]$. In all three lists, the $1$ and $3$ either have a 'distance' of $2$ and $-2$; the same can be said about the $2$ and $4$. Intuitively this makes sense, since this 'distance' is invariant under the two generators $(13)$ (flipping, which negates but doesn't change the gap) and $(1234)$ (shifting also doesn't change the gap). In general it seems to be that for $H := \langle (1,k), (1,...,n) \rangle$, the distance between $1$ and $k$ (and $2$ and $k+1$, etc.) stays constant modulo $\gcd(k-1,n)$.
How do I more formally describe this 'distance' concept? As in, is there specific terminology to deal with this?
Your permutation group is imprimitive, preserving the partition $\{1,3\},\{2,4\}$. This means that every element in this group sends each part into another part. Indeed, it is enough to check that the generators preserve the partition: $(1\;3)$ sends $\{1,3\}$ and $\{2,4\}$ to themselves, whereas $(1\;2\;3\;4)$ switches them.