We study the degrees of freedom (DoF) of the -user interference channel with coordinated multipoint (CoMP) transmission and reception. Each message is jointly transmitted by successive transmitters, and is jointly received by successive receivers. We refer to this channel as the CoMP channel with a transmit cooperation order of and receive cooperation order of . Since the channel has a total of transmit antennas and receive antennas, the maximum possible DoF is equal to . We show that the CoMP channel has DoF if and only if . The key idea is that the zero forcing of the interference corresponding to the message at the decoder of the message, where , can be viewed as a shared responsibility between the transmitters carrying the message, and the receivers decoding the message. For the general case, we derive an outer bound that states that the DoF is bounded above by . For the special case with only CoMP transmission, i.e, , we propose a scheme that can achieve DoF for all , and conjecture that the result holds true for all . In the proposed coding scheme, the transmitters carrying each message are used to cancel the interference introduced by this message at the first receivers, thereby allowing each of these receivers to enjoy 1 DoF, and asymptotic interference alignment is used to align the interfering signals at each other receiver to occupy half the signal space. The achievability proofs are based on the notion of algebraic independence from algebraic geometry.