How rigid compared to what? Normally in situations like this, absent any other information about loads, I would take the normal/recommended structural steel and compute it's characteristics, and then compare that to the type of steel I want to use. That way you get a quantitative answer as to how it compares.
So, what would they normally use? Take that shape, compute the area moment of inertia, and then compare to the area moment of inertia of 1x4 bar.
If loaded against the 1" dimension, the area moment of inertia of a 1x4 bar is going to be 4*1^3/12 = 0.333 in^4. If loaded against the 4" dimension, it would be 1*4^3/12 = 5.333 in^4.
Now make the area moment of inertia calculation for the standard shape normally used, and decide if your 1x4 will cut it. Maybe you'll find out you need a 1.5x4 or something just a little larger, who knows.
I don't know exactly what they use but it looks like two pieces of 4" angle aluminum butted together maybe 5/16" or 3/8" thick.
Using your .333 moment and a 12" cantilever the 1" x 4" bar would deflect 0.3" with a 5,000 lb load, 0.6" with a 10,000 lb load.
I am no engineer but it seems to me that I beam in the first picture sticking up 4' pulled sideways from the top would be a lot weaker than a the fact bar only sticking up 12" ? I don't know.