How does German income tax actually work? Are there bands like the UK?
No — Germany's Einkommensteuer doesn't use flat-rate bands where a fixed percentage applies within each range. Instead, §32a EStG defines a continuous mathematical formula split into five zones: no tax up to €12,348 (the Grundfreibetrag), then two "linear-progressive" zones where the marginal rate climbs smoothly from 14% to 42% as income rises, then a flat 42% zone (the Spitzensteuersatz) from €69,879 to €277,825, then a flat 45% zone (the Reichensteuer) above that. Your marginal rate increases gradually with every euro earned rather than jumping at fixed thresholds.