We develop a technique for ``partially collapsing'' one Markov processes to produce another. The state space of the new Markov process is obtained by a pinching operation that identifies points of the original state space via an equivalence relation. To ensure that the new process is Markovian we need to introduce a randomised twist according to an appropriate probability kernel. Informally, this twist randomises over the uncollapsed region of the state space when the process leaves the collapsed region. The Markovianity of the new process is ensured by suitable intertwining relations between the semigroup of the original process and the pinching and twising operations. We construct the new Markov process, identify its resolvent and transition function and, under some natural assumptions, exhibit a core for its generator. We also investigate its excursion decomposition. We apply our theory to a number of examples, including Walsh's spider and a process similar to one introduced by Sowers in studying stochastic averaging of PDE.