This article pertains to the classification of pairs of simple random curves with conformal Markov property and symmetry. We give the complete classification of such curves: conformal Markov property and symmetry single out a two-parameter family of random curves—Hypergeometric SLE—denoted by $$\mathrm {hSLE}_{\kappa }(\nu )$$ for $$\kappa \in (0,4]$$ and $$\nu <\kappa -6$$. The proof relies crucially on Dubedat’s commutation relation (Commun Pure Appl Math 60(12):1792–1847, 2007) and a uniqueness result proved in Miller and Sheffield (Ann Probab 44(3):1647–1722, 2016). The classification indicates that hypergeometric SLE is the only possible scaling limit of the interfaces in critical lattice models (conjectured or proved to be conformally invariant) in topological rectangles with alternating boundary conditions. We also prove various properties of $$\mathrm {hSLE}_{\kappa }(\nu )$$ with $$\kappa \in (0,8)$$: continuity, reversibility, target-independence, and conditional law characterization. As by-products, we give two applications of these properties. The first one is about the critical Ising interfaces. We prove the convergence of the Ising interface in rectangles with alternating boundary conditions. This result was first proved by Izyurov (Commun Math Phys 337(1):225–252, 2015), and our proof is different. Our method is based on the properties of $$\mathrm {hSLE}$$ and is easy to generalize to more complicated boundary conditions and to other models. The second application is the existence of the so-called pure partition functions of multiple SLEs. Such existence was proved for $$\kappa \in (0,8){\setminus } {\mathbb {Q}}$$ in Kytola and Peltola (Commun Math Phys 346(1):237–292, 2016), and it was later proved for $$\kappa \in (0,4]$$ in Peltola and Wu (Commun. Math. Phys. 366(2):469–536, 2019). We give a new proof of the existence for $$\kappa \in (0,6]$$ using the properties of $$\mathrm {hSLE}$$.