This isn't going to be easy to follow without an assembly at hand and in front of you.
Having purchased and received a pair of replacement clips (though only one of mine was actually broken), I began to figure out how to put the assembly back together not having recalled how I'd taken it apart. It was a bit of a Chinese puzzle that I solved with some trial, error and patience.
The 1st ah-ha moment came when I realized that the orientation of the rack was constrained by its dimensions, the spacing of the rails in the large bottom pull out rack and the placement of the clip flanges. I hadn't remembered that the longer dimension of the assembly paralleled that of the back of the rack. The last factor to consider was the flange on the clips which had to be toward the inside of the assembly.
The final challenge was to make sure the tines of each side were down (the more compact arrangement of the assembly) but over not under the respective rails around which each side rotated.
NOTE: The need for this repair manifested itself because one of the molded plastic ends that functioned as a clip had broken. There are three of these built-in clips on each part. They can easily be broken if care is not taken in using them. Indeed, I suspect that they become more brittle with the use of the dishwasher and, in fact, I ended up using both new parts though I had ordered the second as a back up.
Here's hoping this helps another DIYer.
P.S. Though I checked the use of a screwdriver below, no tools were used. The site wouldn't take my story without checking a tool.