Taper is not the only issue.! Before we get starte on it::

• There were spindles with different offset for single, double and triple cranksets.

• Spindles secured with nuts rather than bolts could not be removed with the crank pullers for older Stronglight or TA cranks.

So, to be sure of a good match, it was necessary to replace a spindle with the same brand and model, or one that was a close match, or have a selection of spindles to try. And then, different spindles would require different bottom bracket cups so the bearing races would fit and the spacing of the cups would be correct. Sutherland's Handbook up through the 6th edition has long tables giving dimensions.  We have a page of them on this site too, though it doesn't cover cross-brand compatibility. And now many of the older spindles and cups are no longer made. Phil Wood cartridge bottom brackets will work with almost anything, and are the ticket when  replacing older spindles.

But OK, tapers: traditional cotterless cranksets use a tapered square spindle (axle) end. The sides of the end taper at a 2 degree angle compared to the centerline, 4 degrees compared to each other.

While there is general agreement about the angle, there is no such agreement about the actual width cut off. There are two standards in current use, and there is another used on some parts still seen. The differences can look subtle, until you actually insert a spindle into a crank.

The spindle approaches the bottom of the dustcap hole in this crank.
There is a risk of the fixing bolt's bottoming out on the spindle.

This crank is at risk of bottoming out on the end of the tapers.

This combination is safe from bottoming out.

The photos below are of Phil Wood cartridge bottom-bracket spindles, which are the same except for the tapers. Note that the shafts of the spindles, at the left side of the photos, are all the same size. The appearance of the tapers on other spindles may differ.

ISO crank on J.I.S. spindle -- small square-taper hole in crank
places it farther outboard on the spindle.

Conversely, if you install a J.I.S. crank on an ISO spindle, it will wind up about 4.5 mm farther in than it would on a J.I.S spindle of the same length.

Theoretically, ISO cranks should only be used on ISO spindles, and J.I.S. cranks only on J.I.S. spindles.

In practice, you can very often get away with mixing these sizes, as long as you select a spindle length that gives the desired chainline.

Taper matching was fairly important back in the day of loose-ball cup-and-cone bottom brackets, because these required regular maintenance/overhauls, and this required removal of the cranks. Every time you remove and re-install a square-taper crank, the hole in the crank is liable to get very slightly larger.

This was particularly an issue when using J.I.S. cranks on ISO spindles, because over time, as the crank went on farther and farther, you could run out of taper, and the square end of the spindle would become flush with the surface the crank fixing bolt/washer pushed against. At that point, further tightening of the bolt won't make the crank any tighter, since the bolt is bumping onto the end of the spindle.

Tolerances on older spindles may not be closely held, and dimensions may vary enough even with the same model number to affect crank/spindle compatibility.

If the end of a bolt-type (hollow, internally-threaded) spindle is too small, it may be ground down, as long as the spindle does not bottom out on the back of the crank and the spindle's internal threads are long enough to hold the bolt.

I (Sheldon) generally avoid mixing sizes on customers' bikes, but I have a lot of experience mixing ISO/J.I.S. in both directions on my own personal bikes, and it has never given me a lick of trouble.

• they are available in all three tapers, different offsets  and several lengths,
• their right-left position is adjustable,
• mounting rings are available for all common threadings.