Pulsars are famed for their rotational clocklike stability and their highly repeatable pulse shapes. However, it has long been known that there are unexplained deviations (often termed timing noise) from the rate at which we predict these clocks should run. We show that timing behavior often results from two different spin-down rates. Pulsars switch abruptly between these states, often quasi-periodically, leading to the observed spin-down patterns. We show that for six pulsars the timing noise is correlated with changes in the pulse shape. Many pulsar phenomena, including mode changing, nulling, intermittency, pulse-shape variability, and timing noise, are therefore linked and are caused by changes in the pulsar's magnetosphere. We consider the possibility that high-precision monitoring of pulse profiles could lead to the formation of highly stable pulsar clocks.