We investigate the electronic and magnetic properties of the proposed one-dimensional transition metal (TM = Sc, Ti, V, Cr, and Mn) -benzene (Bz) sandwich polymers by means of density functional calculations. [V(Bz)]infinity is found to be a quasi-half-metallic ferromagnet, and half-metallic ferromagnetism is predicted for [Mn(Bz)](infinity). Moreover, we show that stretching the [TM(Bz)](infinity) polymers could have dramatic effects on their electronic and magnetic properties. The elongated [V(Bz)](infinity) displays half-metallic behavior, and [Mn(Bz)](infinity) stretched to a certain degree becomes an antiferromagnetic insulator. The possibilities to stabilize the ferromagnetic order in [V(Bz)](infinity) and [Mn(Bz)](infinity) polymers at finite temperatures are discussed. We suggest that the hexagonal bundles composed by these polymers might display intrachain ferromagnetic order at finite temperatures by introducing interchain exchange coupling.