David,
You could use custom geographic coordinate systems (GCS-es) and set of custom transformations (GT-s) to rotate your raster. See the workflow below.
NOTE: Described workflow is not recommended for high accuracy data and ellipsoids, and it does not work well with vector data.
Suggested workflow
To generate a map the globe gores centered on 100° W 40° N, you need to rotate your center point to the North Pole. One way to accomplish that would be using two custom GCS-es and two custom GT-s:
1) Open coordinate system picker, create and store in Favorites two custom GCS-es: e.g. RotatedGlobe1 and RotatedGlobe2.
2) Create a custom geographic transformation from GCS of your data, e.g. WGS 1984, to RotatedGlobe1 GCS. In my example, I used Coordinate Frame method, which includes rotation parameters, passed in seconds: -100° = -360 000". You could also use Longitude Rotation method in this step.
This rotates your raster around Earth's rotation axis, or Z-axis and shifts everything to 100° W meridian.
3) Create another custom geographic transformation from RotatedGlobe1 GCS to RotatedGlobe2 GCS. This time you rotate around Y-axis for 50° = 180 000". This moves your point, 100° W 40° N, to the "North Pole" location.
4) You go back to map properties and select map's coordinate system to be RotatedGlobe2 GCS and select two custom transformations to rotate your data properly.
Here is the resulting rotated raster:
5) You can also create a custom polyconic projected coordinate system and automatically project your data.
Note: In ArcGIS Pro 2.5, there are some issues with reading custom coordinate systems from Favorites, so you might have to manually input correct GCS definition when creating it. If done correctly, it will pick your custom transformations.
Here is the result.
6) You might want to overlay with un-transformed graticule to properly clip a gore. Create a copy of 15° graticule vector layer, use Define Projection tool to change GCS of that layer to be RotatedGlobe2 GCS, and add it to your map.
7) To generate other gores, just change Central Meridian parameter to shift everything to the center of the next gore.
Again, you might have to manually input correct GCS definition when creating it due to a bug in ArcGIS Pro 2.5.