After thinking about the geometry involved for this particular type of ambigram, I understood that I was aiming to make a word that would look the same under a 180 degree rotation. With this in mind, I tried to think of words whose first and last letters might bear some similarities under rotation. Obviously, every letter would have to have its rotated partner, but I had to start (and end) somewhere. Geometry seemed like it could work well. It wasn't too much of a stretch to imagine a capital G that looked something like a y.
Having settled on a word to try, I listed out the pairs of letters that would have to be rotations of one another. In retrospect, I suppose the best ambigrams don't necessarily match one letter with one other letter, but carefully weave shapes and features even between "letters" to achieve the desired effect. To keep it simple for myself, though, I tried to work with one letter pair at a time: G and y, e and r, o and t, m and e.
I filled up the margins of my notepad with sketches of my attempts at making symbols for these pairs. I basically tried to write each letter with the other in mind, and then I would rotate my notebook around 180 degrees and add more marks to the letter to make it look like the other one. There were lots of attempts that I didn't use. The challenge felt like walking a fine line between ambiguity and specificity in what letter(s) each symbol represented.
I had the hardest time with e and r. I was tempted to just borrow Scott Kim's solution to this one in his "SuperTeacher" inversion, but I kept trying and came up with something else that worked. My solution to m and e, right in the middle of the ambigram, was to go with a basic lowercase m and turn it 45 degrees. Not fancy, but again, it worked. Once I had each symbol figured out (four of them altogether, since it was two letters for each), I set about drawing a large copy of the ambigram.
To draw it accurately, I decided to mark the center of a piece of paper and then to draw out half of the word (the first, and the last half :) ) on one side of this center. Then I used an architect's scale to make a connect-the-dots copy of this half as a reflection through the center point. The resulting point symmetry is the same thing as 180 degree rotation symmetry. This was tedious, keeping the ruler lined up through the center and mirroring distances for lots of points. While I worked I kept thinking of more efficient ways to do this, like scanning the image and rotating it on the computer, or photocopying it, or even just tracing it on another piece of paper and attaching the two. But for the sake of having my finished product completely hand drawn, and on one, intact piece of paper, I persisted.
The finished product turned out fairly well. Here are two photographs of the completed ambigram. I lined up the corners of the paper with the ruler and the pencil the same way in each, but the paper does get rotated 180 degrees. Fairly similar?