Consider the grid paths i.e. paths made up of $(1,0)$ and $(0,1)$ steps that go from $(0,0)$ to $(n,n)$. There are ${2n \choose n}$ of these paths. Now consider the subset of paths $P_k$ for $k=1,...,n$ such that the first $2k-1$ steps consists of at least $k$ 'right' steps. In particular, $P_1$ is just the paths that start with a 'right' step. By symmetry, $P_k$ consists of half of all paths. My question is if there are natural bijections that will take me from $P_k$ to $P_{k'}$. I'm particularly interested in a bijection between $P_k$ and $P_1$.
One choice would be to reflect the paths that are in $P_k$ and not in $P_{k'}$ through the diagonal $x=y$, but this is not 'natural' since it requires me to separate paths into two groups.