The winter polar vortex on Saturn's largest moon Titan has profound effects on atmospheric circulation and chemistry and for the current northern midwinter season is the major dynamical feature of Titan's stratosphere and mesosphere. We use 2 years of observations from Cassini's composite infrared spectrometer to determine cross sections of five independent chemical tracers (HCN, HC₃N, C₂H₂, C₃H₄, and C₄H₂), which are then used to probe dynamical processes occurring within the vortex. Our results provide compelling evidence that the vortex acts as a strong mixing barrier in the stratosphere and mesosphere, effectively separating a tracer-enriched air mass in the north from air at lower latitudes. In the mesosphere, above the level of the vortex jet, a tracer-depleted zone extends away from the north pole toward the equator and enrichment is confined to high northern latitudes. However, below this level, mixing processes cause tongues of gas to extend away from the polar region toward the equator. These features are not reproduced by current general circulation models and suggest that a residual polar circulation is present and that waves and instabilities form a more important part of Titan's atmospheric dynamics than previously thought. We also observe an unexpected enrichment of C₄H₂ in the northern stratosphere, which suggests photochemical polymerization of C₂H₂. Our observations provide stringent new constraints for dynamical and photochemical models and identify key polar processes for the first time. Some of the processes we see have analogues in Earth's polar vortex, while others are unique to Titan.