Normally, linear crossed-dipole arrays (LCDA) can only perform 1-D direction of arrival (DOA) estimation. In this article, a prior known mirror reflection structure is added for LCDA and a mirrored LCDA (MLCDA) capable of 2-D DOA estimation is constructed. The joint DOA and polarization parameter estimation performance of MLCDA is significantly improved due to the added mirror reflection structure. First, it is proven that an MLCDA with half wavelength interelement spacing (IES) can perform ambiguity-free joint 2-D DOA and polarization estimation when the array is neither parallel nor perpendicular to the added mirror reflection structure. Then, the closed-form IES boundary for ambiguity-free 2-D DOA and polarization estimation is derived. The parameter estimation accuracy and multitarget resolution can be further enhanced by increasing the IES under such a boundary. Moreover, a computationally efficient compressed mirrored dimension-reduction multiple signal classification (C-MDR-MUSIC) algorithm for MLCDA is proposed. Initially, a dimension-reduction processing for the manifold vector of MLCDA is achieved. Then, the range of spatial spectrum search required for MDR-MUSIC is halved, employing the conjugate symmetry property between the spatial steering vectors of the incident and reflected signals. Numerical simulations are conducted to validate the performance of MLCDA and the proposed C-MDR-MUSIC algorithm.