For the record, I posted before reading Gordo's response, and I'm reluctant to argue against his advice. But...

... with euclidean's checkerboard pattern, the elongated axis incurs the tone bump first, and the second axis occurs a bit higher up in the tone scale, making this not such an abrupt transition, no?
No.
I think you're confusing Euclidean with Elliptical.
Euclidian is round/square/round(inverted) The dot shapes are non-directional - there is no elongated axis.
Elliptical dot shapes are diamondish ellipticalish. They have an elongated axis.
Here's what Euclidean looks like close up. This sample is Agfa Balanced Screening 150 lpi Euclidean dot (it's from some screening work I did with them a few years ago)
The big problem with Euclidean, as I mentioned, is that the round dots become square - a checkerboard - when they go through the 50% tone. This causes an "optical bump" - a visible dark line at the 50% tone - when the four corners of the dot all touch at the same time. It's especially visible in vignettes.
BTW - Agfa uses a neat trick to try and minimize this issue. You can see it if you examine the black printer up close. (ignore the extra color dots you'll see they are caused by a color management error)
Elliptical mitigates this by splitting the point where dots touch to the 40 and 60% tone value - but introduces other issues - some of which I mentioned before.
Round dots put the optical bump at the 75% tone - i.e. deep in the shadows where it is less visible.
Here's a link to an animation of an elliptical screen to make it easier to understand:
http://qualityinprint.blogspot.com/2008/12/150-lpi-elliptical-halftone-dot.html
RE: "Also, the checkerboard pattern is a dead givaway to the true 50%"
To form a perfect checkerboard at 50% requires that the halftone frequency (lpi) be an even divisor of the dpi of the imaging device as well as being image with a non-gaussian laser - which seldom happens in the real world.
Glad you're using FM screening though!
and PeterA wrote:
About 4 years ago I had someone from Creo arrive at my company who said he developed the square spot and did a trick with a square cut out of a piece of paper and a lighter - I cant remember what he was demonstrating though
That person was likely Dan Gelbart and was trying to demonstrate SquareSpot imaging technology - not halftone screening. Laser spot vs halftone dot. You're not the only one who had a difficult time understanding the difference - or its significance.
best, gordo