In one-sided matching with strict ordinal preferences, the Top Trading Cycles (TTC) mechanism maps every initial endowment to a Pareto-optimal (PO) allocation, and many distinct endowments converge to the same outcome. We study the \emph{inverse} problem: given a PO allocation $σ$, characterize and enumerate its preimage $f^{-1}(σ)$, the set of endowments from which TTC produces $σ$. We characterize $f^{-1}(σ)$ via ordered partitions of the agents and present a sound and complete enumeration algorithm, invTTC. The preimage carries direct economic meaning: $|f^{-1}(σ)|/n!$ equals the probability that random serial dictatorship selects $σ$, and $f^{-1}(σ)$ is the basin of attraction of $σ$, quantifying robustness to the assignment of initial property rights. We analyze the enumeration's complexity and report experiments on preimage structure.

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