Will the Garmin Edge Touring store an in progress ride if power is lost? A rider is on a long ride, using a Garmin Edge Touring GPS unit and takes a break. If they wish to save battery on the Garmin, can they power it off and continue the same ride later?

I believe the Edge Touring is the same as the 810, with some features turned off.
Yes

A comment below made me wonder could a planned turn off be different to a random turn off like my original answer below.

So a quick test this morning and... 
if you turn it off through the button on the side when you turn it back on you will have the Discard and Save option. 
Press home, navigate to the trip screen again and your ride is ready to resume, just press the start button again.

If the Garmin decides to turn itself off though the data stored before that is lost.



Old answer... Proved to be incorrect 

Not without some manual steps

You will need to manually stop the ride by pressing the start/stop button, then save the ride from the options presented, before turning of the Garmin. 

Then turn the Garmin back on before riding off. 

You will then be able to stitch the GPS files together to make one continuous journey if you don't want your ride broken into multiple sections

If the power turns off mid ride that section of the ride is lost. 

I know this because my Touring turned itself off a few weeks ago mid ride and I lost the first 5km of my ride.