Should you do this with a computer at all?

I only sit down in front of the computer once the design is done on paper. I've tried making Ambigrams on the computer directly, but found it too cumbersome compared to paper. However, re-drawing a design on the computer will often result in a nicer and more consistent lettering and can help making the words really symmetric.

For most of the Ambigrams above, I used the Inkscape vector graphics editor. It is free software and available for Linux, Windows and Mac OS, so it may also be the right tool for you. :)

Somewhere in his Gallery, Scott Kim says that he uses Adobe Illustrator, so if you already own that, that might work for you too.

Some Inkscape tricks

• Align using the grid: The grid seems to be more useful for alignment than guides. I haven't found a good way to mirror the guides in a symmetric way, but with the grid it's easy to make things symmetric by counting boxes.
• Reuse parts of your letters: It can be useful to split your letters into several reusable pieces that you can copy&paste individually. That way, you can reuse the same serifs and curves in different places, which gives the letters a much more finished and consistent feel. You can later join these pieces again with the “Union” operation (see below).
• For symmetry, use clones: Start with the left side of the word, then:
  1. Start with the left side of the object (Bezier tool, “Edit Nodes” tool)
  2. Select all objects
  3. Use “Edit” > “Clone” > “Create Clone” to clone them all.
  4. Move the cloned objects to the right position.
  5. Turn the cloned objects on their head.
     • In my Inkscape version, “Rotate selection” did the wrong thing and ended up rotating each of the objects around its own center, instead of a common center.
     • To rotate around a common center, select all cloned objects and click the selection to get the rotation handles. Rotate the selection with the rotation handles while pressing the Ctrl key. Pressing the key will make the rotation snap at multiples of 45°.
  6. Continue tweaking the original objects with the “Edit Nodes” tool. The clones will reflect the changes directly, and you only sometimes need to move them a little to keep both sides in sync.
• Coloring: I do this late during editing. It's often helpful to use “Path” > “Union” before applying fill and stroke, so that you really only have the stroke surrounding the letters. You'll need to un-link the clones for this to work (“Edit” > “Clone” > “Unlink Clone”). I usually store these changes only into a new file, so that I can go back to the version where the clones are linked.