say $t_1$ is *equivalent* to $t_2$ if ${\operatorname{{\mathbb O}}}_{t_1} = {\operatorname{{\mathbb O}}}_{t_2}$ and there is an orientation-preserving homeomorphism $\phi \in {\operatorname{Mod}}(S_g)$ such that

(i) $\phi(P_{1,i}) = P_{2,i}$ for $1 \leq i \leq r$,

(ii) for each $1\leq j\leq k$, if ${\operatorname{{\mathbb O}}}_{s,j} = \{Q_{j,1}^s, Q_{j,2}^s, \ldots, Q_{j,m_{s_j}}^s\}$, then $m_{1_j} = m_{2_j}$ and $\phi(Q_{j,i}^1) = Q_{j,i}^2$ for all $1\leq i\leq m_{1_j}$, and

(iii) $\phi t_1 \phi^{-1}$ is isotopic to $t_2$ relative to ${\operatorname{{\mathbb P}}}(t_2)\sqcup(\cup_{j=1}^k {\operatorname{{\mathbb O}}}_{2,j})$.

The equivalence class of a permuting $(n,r,k)$-action is denoted by ${\operatorname{{\llbracket}}}t{\operatorname{{\rrbracket}}}$.

We now introduce the notion of an *$(n,r)$-data set*, which encodes the signature of the quotient orbifold of a permuting $(n,r,k)$-action and the turning angles around its distinguished fixed points. Furthermore, the $(n,r)$-data set will be combined with the orbit distribution of the action to form a pair, which we will call a *permuting $(n,r,k)$-data set*.

\[defn:data\_sets\] Given $n\geq 1$ and $r\geq 0$, an *$(n,r)$-data set* is a tuple $${\operatorname{{\mathcal D}}}= (n,g_0, (a_1,a_2,\ldots, a_r); (c_1,n_1), (c_2,n_2),\ldots, (c_s,n_s))$$ where $n\geq 1$ and $ g_0 \geq 0$ are integers, each $a_i$ is a residue class modulo $n$, and each $c_i$ is a residue class modulo $n_i$ such that:

(i) each $n_i\mid n$,

(ii) $gcd(a_i,n) = gcd(c_i,n_i) 