We show that the space symmetry of bilayer systems, coupled by both tunneling and Coulomb interaction, has fundamental influences on the collective charge-density excitations. In the general case of two-subband occupation, three modes with one intersubband and two (optical and acoustic, respectively) intrasubband plasmons are calculated. By breaking the space symmetry, these modes are found to be mixed, giving rise to an anticrossing behavior between the intersubband and optical intrasubband plasmons. For the asymmetric case, the acoustic intrasubband plasmon is not Landau damped in the long-wavelength limit.