These notes introduce a new class of algebraic curves on Hilbert modular surfaces. These curves
are called twisted Teichmüller curves because their construction is very reminiscent of
Hirzebruch-Zagier cycles. These new objects are analyzed in detail and their main properties
are described. In particular the volume of twisted Teichmüller curves is calculated and their
components are partially classified. The study of algebraic curves on Hilbert modular surfaces
has been widely covered in the literature due to their arithmetic importance. Among these
twisted diagonals (Hirzebruch-Zagier cycles) are some of the most important examples.
