We report the crystal structures of the novel technetium tetrahalides TcX4 [X = F, I], as predicted from first-principles calculations. Isomorphous with TcCl4 and TcBr4 crystals, TcF4 is orthorhombic with the centro-symmetric space group Pbca, while Tcl(4) crystallizes in the monoclinic space group P2(1)/c. The structures, [TcX2(mu-X)(4/2)](infinity), consist of distorted edge-sharing octahedral groups of composition TcX6 linked into endless cis chains. A possible explanation for the differences between these structures is offered in terms of varying degrees of bonding within the chains.