So the new 4x5 that I bought has base tilts, which I've never had before

[Bill Koechling]

I did a lot of product photography with an old axis-tilt Calumet 4x5 and Cambo 8x10. The clients liked to have every part of each product in focus. Tilts, swings, rise or falls and every possible adjustment - many times extreme - was sometimes required to meet that standard. When I got a base-tilt Arca Swiss focus was easier to maintain. That's really about it. For landscape and most portraits it's pretty much a non issue.

[Ivan J. Eberle]

From an engineering standpoint, some of the axis tilt knobs are ridiculously simple, being not much more than a small bolt and knob. It's easier and cheaper to design since there's not as much clamping force needed as
with an off-axis tilt.
Conversely, a camera like a Sinar Norma, with base tilts, requires a beefier and more complicated clamping arrangement, and a lot more oomph applied to it to keep the standard from creeping with a heavy lens or a lot of front tilt applied.

[Rod Klukas]

Axis tilt: Focus near, tilt far

Base tilt: Focus far, tilt near.

With base tilt with a very short focal length it is easier to tilt the front, or the back, as you will have less mechanical conflict with the standards or bellows bunching up.
Rod

[Leigh]

Back tilt changes the geometry of the subject.

Front tilt does not.

- Leigh

[Ivan J. Eberle]

Well, you can always straighten the geometry back out in Photoshop

[Leonard Evens]

My Toho FC-95X has base tilt and axial swing, so I am familiar with both. In either case, I proceed by starting with a guess, usually about 5 degrees, and then choose two points in my desired exact plane of focus, focus on the far one and then focus on the near one. If that requires bringing the standards further apart, I increase the tilt and if it requires separating the standards more, I decrease the tilt. Usually a couple of iterations gets the exact plane in focus. I don't see much advantage with axial tilts when I do it this way.

[mcherry]

Thanks for all the responses guys! I never thought there would be so many to an apparently often asked about question. I've been having a ton of fun playing with the movements and this weekend plan to do some shooting with it. I instinctively reach for the axial tilt, as that's what I know, but since reading all of these have decided to focus on working with the base tilts to get a handle on those.

So much fun!