Snapshot hyperspectral imaging avoids sequential scanning, but systems that jointly achieve stable reconstruction, low cost, and compact optics remain limited. We present a snapshot hyperspectral imaging system based on angular-to-spectral diversity conversion. A tapered kaleidoscope creates replicated views with distinct incidence directions, and a directly attached birefringent filter converts them into view-channel-dependent spectral transmittances, yielding complementary measurements that better condition the inverse problem for more stable single-shot spectral reconstruction. The system preserves a simple pixel-wise linear model for fast non-learning-based reconstruction and uses only off-the-shelf components without relay optics or cascaded modules. We select the birefringent filter configuration using a condition-number-based criterion and validate the system on both synthetic and real data.