Ambigrammatic - Natural Glyphs
You can start making your own ambigrams without even having to design anything new.
Glyphs are the basis of all writing, and they are also the basis of ambigram construction. There are many glyphs that are naturally symmetrical and can be used as the foundation of your design.
O - The ultimate ambigram glyph, an O can be reflected and rotated all you want and you still have an O. Perfect.
X - Rotates itself at 90 degree rotations, and reflects itself horizontally and vertically. Also might be used as a plus sign or "t" when rotated 45 degrees, but that probably isn't too practical.
I - Almost as good as O and X. Reflects horizontally and vertically. Rotated 90 degrees can become an H if the serif is wide enough. Sans-serif, can also act as a lower case L, or a one.
H - See I. Reflects horizontally and vertically.
M and W - Rotations of each other, they have horizontal reflective symmetry, and reflect each other vertically. Excellent. Rotated 90 degrees, they become a 3 or an E. Therefore 3 and E also share the same reflective relationship as M and W.
N and Z - Rotations of each other at 90 degrees, and of themselves at 180 degrees.
S - 180 degree rotation of itself.
d p q b - d is a rotation of p, q is a rotation of b, while d is a horizontal reflection of b, and p is a horizontal reflection of q AND d is a vertical reflection of q, and b is a vertical reflection of p. Tons of fun!
u and n - 180 degree rotations of each other. Could also be rotated 90 degrees for a ca and e - in many fonts these are 180 degree rotations of each other.
T - Horizontal reflective symmetry.
t and f - vertical mirrors of each other, at least in most serif fonts. You may have noticed that the tumblr and facebook logos are perhaps a bit too similar to each other.
j and r - Somewhat rotations of each other, with an easily overlooked dot.
V and A - Also somewhat rotations of each other.
6 and 9 - Probably the most well known rotational symbiotograms out there.
2 and 5 - In some fonts these are exact horizontal/vertical reflections of each other, and both are rotations of themselves.
L and 7 - Rotations of each other.
8 - 180 degree rotation of itself, and reflects itself horizontally and vertically.
And then you can start getting into the number-letter replacement of leetspeak to add 4=A, 7=T (and therefore L rotates as T), 5=S, and so on.
I am probably forgetting a few examples, but that should be enough to get anyone started playing with all the different kinds of ambigrams that are possible. The next step is to add slight modifications that keep a glyph readable, yet alter how a glyph is read when reflected/rotated. Stay tuned for more tips and tricks for ambigram construction!