I found two references that I think lock in the most likely answer at 60 cylinders, but a close reading of the definition, could reasonably lead to an interpretation of 30, 60, or 120.
If one ignores the potential for flipping the opposite side map (antipodal?) then 30 cylinders could be the smallest number, as mkennedy spelled out above.
The 60 cylinders model is the most common approach used to visualize them, but interestingly enough, the two references only mention 120 Spatial Reference Frames or instances.
First up, the US National Geospatialintelligence Agency (NGA) defines UTM as follows:
UTM is a family of 120 instances of the general form of the transverse
Mercator projection. Each instance is called a zone and is given a
zone number Z between -60 and +60 excluding zero.
Source:NATIONAL GEOSPATIAL-INTELLIGENCE AGENCY STANDARDIZATION DOCUMENT Implementation Practice The Universal Grids and the Transverse Mercator and Polar Stereographic Map Projections 2014-03-25, NGA.SIG.0012_2.0.0_UTMUPS, Section 7.1 page 32.
Second, the International Standards Organization (ISO) group has published the standard for Spatial Reference Models, and it defines UTM as:
A set of 120 localized Spatial Reference Frames (SRFs),
where limited overlap is modelled by extended validity regions in the member SRFs.
Source: International Standard ISO / IEC 18026, Information Technology: Spatial Reference Model, ISO / IEC 18026, Section 8.7.7 - Table 8.6.0, page 192.
So, one could also make a case for 120 cylinders (half cylinders?) since each UTM zone could be aligned to a unique one. If the mappers use different ellipsoids for their UTM projections, then the cylinders will naturally be different too.
I think the UTM standard for the US to only use the WGS 1984 ellipsoid - and that is the standard. The referenced NGA document also spelled out approaches for TM to be applied to many other ellipsoids, starting at page 20.
I hope this helps - corrections welcome.