We study the sum degrees of freedom (DoF) of the bursty MIMO X channel without feedback, where the four transmitter-receiver links are intermittently on-and-off, controlled by four Bernoulli random sequences which may be arbitrarily correlated, subject to a symmetry assumption: The two direct-links have the same level of burstiness, modeled by $\mathrm{Ber}(p_d)$, and so do the cross-links, modeled by $\mathrm{Ber}(p_c)$. The sum DoF is fully characterized in the regime where $\frac{p_c}{p_d}$ is small, i.e. below a certain threshold, and is partially characterized in the other regime where $\frac{p_c}{p_d}$ is above the threshold. The achievability is proved with a combination of Han-Kobayashi strategy and interference alignment, which can achieve strictly higher DoF than interference alignment alone. The converse proof employs a channel-state-sequence pairing technique. We highlight that burstiness of the channel disrupts the network topology, turning the MIMO X channel into a network with time-varying topology. This fundamental difference has striking ramifications. In particular, various interference alignment schemes that achieve the DoF of non-bursty X channels are found to be suboptimal when the channels become bursty. The reciprocity between the forward and the reverse links is lost, and the sum DoF does not saturate when the ratio of the transmitter and the receiver antennas exceeds $\frac{2}{3}$.