A distinguishing $r$-labeling of a digraph $G$ is a mapping $\lambda$ from the set of vertices of $G$ to the set of labels $\{1,\dots,r\}$ such that no nontrivial automorphism of $G$ preserves all the labels. The distinguishing number $D(G)$ of $G$ is then the smallest $r$ for which $G$ admits a distinguishing $r$-labeling. From a result of Gluck (David Gluck, Trivial set-stabilizers in finite permutation groups, {\em Can. J. Math.} 35(1) (1983), 59--67), it follows that $D(T)=2$ for every cyclic tournament~$T$ of (odd) order $2p+1\ge 3$. Let $V(T)=\{0,\dots,2p\}$ for every such tournament. Albertson and Collins conjectured in 1999 that the canonical 2-labeling $\lambda^*$ given by $\lambda^*(i)=1$ if and only if $i\le p$ is distinguishing. We prove that whenever one of the subtournaments of $T$ induced by vertices $\{0,\dots,p\}$ or $\{p+1,\dots,2p\}$ is rigid, $T$ satisfies Albertson-Collins Conjecture. Using this property, we prove that several classes of cyclic tournaments satisfy Albertson-Collins Conjecture. Moreover, we also prove that every Paley tournament satisfies Albertson-Collins Conjecture.