What a very interesting problem to solve.
IF your only talking a few thousand shapes I have a simple solution.
Only have your polygons in the map
Set up data driven pages with your polygons as the layer. Make the outline of your polygons a nice thick layer and maybe even give them random colors for a fill.
When setting up DDP sort by objectID
set up labels for the polygons with objectID right in the middle and big enough to see easily.
Page set up to 8 1/2 by 11 with no margin.
Export the DDP as PDF multiple sheets using a . ( find that tends to work better with large numbers)
use lower resolution and faster output. When they are all created combine the pdf't into a single pdf.
then simply eyeball them. you can use your arrow keys to move from one to the next pdf. when you find one that is shaped like an L or a T then you address that situation.
sounds cumbersome but you should be able to get to the point of browsing the PDF's in very short time, depending on how many polygons your talking. If you lower the resolution and quality and only have the polygons on the map you can generate many thousands of PDF's very quickly.
I have a more complicated mathematical idea in my head but I need to think on that.
I like the scan idea... then use optical scanning with character recognition software... but there is a difference between T and L and T-like-ish. But unless you have to do this many times, cranking up the mouse and doing onscreen selection is going to be way faster regardless of other automated systems
Not an immediate solution to your question, but related - the Remote Sensing world now has a variety of Feature Extraction processes developed to do much of what you want. However, while ArcGIS has some of the functionality, to my knowledge it is missing some of the pieces needed to make this happen without purchasing third-party add-on software (however, users of ArcGIS 10.3 and 10.4 can correct me if ESRI has added to the capabilities).
For example, in a class in Remote Sensing I had we used Feature Extraction processes to automatically identify and extract helicopters in an aerial image full of aircraft on a tarmac.
As another example, here's a feature extraction process that involves ArcMap and eCognition:
Chris Donohue, GISP
Without more information about your dataset this will be a difficult question to answer. Are your shapes all the same size, are they all aligned to some underlying grid, randomly placed, overlapping etc? Edit your question and give more context!
One thing I was thinking was that a T shape has 8 right angle bends whilst an L has 6...
You would have to employ some pattern recognition .... Here is an good write up about going about identifying some basic geometric shapes..... this may get you started in determining what you need if you need real time automation.
Some other initiatives that looked interesting: