Heterostructured two-dimensional colloidal nanoplatelets are a class of material that has attracted great interest for optoelectronic applications due to their high photoluminescence yield, atomically tunable thickness, and ultralow lasing thresholds. Of particular interest are laterally heterostructured core-crown nanoplatelets with a type-II band alignment, where the in-plane spatial separation of carriers leads to indirect (or charge transfer) excitons with long lifetimes and bright, highly Stokes shifted emission. Despite this, little is known about the nature of the lowest energy exciton states responsible for emission in these materials. Here, using polarization-controlled, steady-state, and time-resolved photoluminescence measurements, at temperatures down to 1.6 K and magnetic fields up to 30 T, we study the exciton fine structure and spin dynamics of archetypal type-II CdSe/CdTe core-crown nanoplatelets. Complemented by theoretical modeling and zero-field quantum beat measurements, we find the bright-exciton fine structure consists of two linearly polarized states with a fine structure splitting similar to 50 mu eV and an indirect exciton Lande g-factor of 0.7. In addition, we show the exciton spin lifetime to be in the microsecond range with an unusual B-3 magnetic field dependence. The discovery of linearly polarized exciton states and emission highlights the potential for use of such materials in display and imaging applications without polarization filters. Furthermore, the small exciton fine structure splitting and a long spin lifetime are fundamental advantages when envisaging CdSe/CdTe nanoplatelets as elementary bricks for the next generation of quantum devices, particularly given their ease of fabrication.