We give an explicit description of the jointly invariant measures for the KPZ equation. These are couplings of Brownian motions with drift, and can be extended to a process defined for all drift parameters simultaneously. We term this process the KPZ horizon (KPZH). As a corollary of this description, we resolve a recent conjecture of Janjigian, Rassoul-Agha, and Seppalainen, by showing the existence of a random, countably infinite dense set of directions at which the Busemann process of the KPZ equation is discontinuous. This signals instability and shows the failure of the one force–one solution principle and the existence of at least two extremal semi-infinite polymer measures in the exceptional directions. As the inverse temperature parameter for the KPZ equation \beta goes to \infty, the KPZH converges to the stationary horizon (SH), a process which describes the jointly invariant measures for the KPZ fixed point. As \beta approaches 0, the KPZH converges to a coupling of Brownian motions that differ by linear shifts, which is a jointly invariant measure for the Edwards-Wilkinson fixed point. Joint work with Sean Groathouse, Firas Rassoul-Agha, and Timo Seppäläinen.