I am looking for a reference for the following class of linear orderings on a non-zero ordinal $\lambda$.
Define $\lambda^-$ as the predecessor of $\lambda$ if $\lambda$ is a successor, and set $\lambda^-:=\lambda$ otherwise. Let $N:\lambda^-\longrightarrow \{-1,1\}$ be a function.
For $\alpha,\beta \in \lambda$ with $\alpha<\beta$, we set $\alpha<_N\beta$ if $N(\alpha)=1$ and $\beta <_N \alpha$ if $N(\alpha)=-1$. This defines a linear ordering on $\lambda$.
In fact, for all ordinals $\gamma,\eta<\lambda$, we have $\gamma<_N\eta$ if and only if the restriction of $N$ to $\gamma$ is smaller than its restriction to $\eta$ for the lexicographic ordering on functions $\lambda^- \rightarrow \{-1,0,1\}$ (after padding with zeroes on the restrictions so as to extend them to $\lambda^-$). So this ordering is the ordering on the nodes of a path $N$ in a full, lexicographically ordered binary tree with height $\lambda^-$.
Orderings of the form $<_N$ on $\lambda$ are characterised by the following: for all $\alpha<\lambda$, the subset $[\alpha,\lambda) \subseteq \lambda$ is convex for $<_N$.
Let $\mathbf{L}$ denote the class of linear orderings that are isomorphic to a $(\lambda,<_N)$ for given $\lambda$ and $N: \lambda^-\longrightarrow \{-1,1\}$. This class is closed under convex subsets and reverse orderings.
Question: Is the class $\mathbf{L}$ defined and studied somewhere?