There are two semi-connected concepts at play here. Polarization in this context refers to the ratio of neutralizing (i.e. "up" vs "down") spins in a given system. For most nuclei in organic systems like protons, carbons, and nitrogens, this ratio is naturally very small, which is the reason that magnetic resonance approaches like MRI usually have poor signal-to-noise. Hyperpolarization techniques usually involve the transfer of polarization from a source of high ratio, like a free electron, to a relevant target (in the original poster's example, 13C in pyruvate). The polarization in this case is hyperpolarized 13C, which has an "up"-to-"down" spin ratio that is much higher than regular 13C, which makes the signal-to-noise that you get from the pyruvate much higher than it would be otherwise. Tumors love pyruvate so this approach means that tumors will light up like a beacon in your MRI.

The physical rotation/tumbling of molecules in an MRI is also very important, because the strong magnetic field is the thing inducing the "up"-vs-"down" split in the first place, and if the molecular motion is happening at a certain frequency with respect to the external magnetic field there are other interactions that can come into play which can affect the coherence of the nuclear spins (i.e. they can fall out of sync). Thankfully, the rotation of a small molecule like pyruvate is very fast (might higher then the "spin" frequency-a.k.a the Larmor frequenct of 13C at the magnetic field strengths involved in MRI) so the physical tumbling of pyruvate doesn't really come into play when trying to measure its signal. It can be another story for molecules that don't tumble quickly, like the ones that make up tissues, fat, etc.

Great explanation. Indeed, the "rotation/tumbling" is nothing to worry about. One of our biggest challenges is relaxation. The moment we polarize, the clock ticks. Usually, the time scope is around 15-90s. I'm in the field of para-hydrogen and built automated systems which carry out the chemical reactions, polarization transfers and sample cleaning. Many hyperpolarization experiments with para-hydrogen prefer nasty solvents as chloroform or methanol. It is a technical challenge to replace them by water within seconds. One of my favorite topics is Xenon hyperpolarization. It's very elegant, no cleaning required, no wet chemistry, and provides amazing lung scans.