We find that the $^{55}\mathrm{Mn}$ NMR linewidth and Knight shift of icosahedral ${\mathrm{Ti}}_{63}$${\mathrm{Mn}}_{37}$ (i-TiMn) are identical to those of the crystalline phases obtained upon annealing. Comparison of the linewidth in i-TiMn and a bcc Ti-Mn solid solution shows that the Mn sites in both phases have a nearly cubic or higher symmetry. These measurements and our analysis provide strong evidence for a large density of structurally ordered, highly symmetric sites in an icosahedral alloy and demonstrate that a high degree of structural disorder is not an intrinsic property of the icosahedral phase.