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We study the band structure and scattering of in-plane coupled longitudinal and shear stress waves in linear layered media and observe that exceptional points (EP) appear for elastic (lossless) media, when parameterized with real-valued frequency and tangential wave vector component. The occurrence of these EP pairs is not limited to the original stop bands. They could also appear in all mode pass bands, leading to the formation of new stop bands. The scattered energy near these locations is studied along with the associated polarization patterns. The broken phase symmetry is observed inside the frequency bands book-ended by these EP pairs. This is especially manifested by the chirality of the trajectory of the particle velocity, which gets selected by a ‘‘direction’’ of the wave, e.g. the imaginary part of normal component of the wave vector, or the energy flux direction just outside the band. Additionally, EP pairs also appear in the spectrum of the (modified) scattering matrix when mechanical gain is theoretically included to balance the loss in a parity-time symmetric finite structure. These EP pairs lead to amplification of transmission to above 1 and single-sided reflectivity, both phenomena associated with broken phase symmetry, with intriguing potential applications.