We apply the theory of Gröbner bases to the study of signed, symmetric polyomino tilings of planar domains. Complementing the results of Conway and Lagarias we show that the triangular regions $T_N=T_{3k-1}$ and $T_N=T_{3k}$ in a hexagonal lattice admit a \emph{signed tiling} by three-in-line polyominoes (tribones) \emph{symmetric} with respect to the $120^{\circ}$ rotation of the triangle if and only if either $N=27r-1$ or $N=27r$ for some integer $r\geq 0$. The method applied is quite general and can be adapted to a large class of symmetric tiling problems.