The avoided-crossing molecular-beam method has been applied to CH3SiD3 in the ground torsional state (upsilon(6)=0) Three "rotational" anticrossings have been measured corresponding to normally forbidden transitions in which both the rotational and leading torsional energy terms change. Each torsional sublevel with (J=4,k=-/+) and given torsion-rotation symmetry Gamma undergoes an avoided crossing with its counterpart with (J=3,k=+/-2) and the same Gamma. Four "barrier" anticrossings have been measured corresponding again to normally forbidden transitions, but in which only the torsional energy changes. These transitions are (J<->J), (k=+/-1 <-> -/+1), and (Gamma = E-3 <-> E-2,E-1) for J=1 and 2. From these seven zero-field splittings and nine existing R-branch microwave frequencies for upsilon(6)less than or equal to 2, nine torsion-rotation parameters have been determined including the effective rotational constant A(eff) = 34 192.04(11) MHz and the effective height of the barrier to internal rotation V-3(eff)=585.08(5) cm(-1). For each anticrossing studied, an estimate has been made of the contribution delta nu(hyp) to the zero-field splitting from the nuclear hyperfine interactions. For CH3SiH3, CH3CD3, and CH3SiF3, barrier anticrossings have been previously investigated. For each of these anticrossings, estimates of delta nu(hyp) are made here as well. For all cases studied (including those for CH3SiD3), it is found that \delta nu(hyp)\ less than or similar to 5 kHz. For CH3SiD3, by using conventional electric-resonance molecular-beam methods, the electric dipole moment has been determined to an accuracy of similar to 55 ppm for each of the rotational states (J,k)=(1,+/-1), (2,+/-1), and (3,+/-2).