Some letters of the alphabet can easily be formed so that they have symmetry. Letters AMQTUVWY can be formed so that they are symmetrical over a vertical line. The right half of the letter is a mirror image or a reflection of the left half of the letter. If the shape is flipped horizontally, it appears the same.
Letters BCDEK can be formed so that they are symmetrical over a horizontal line. The top half of the letter is a mirror image or a reflection of the bottom half of the letter. If the shape is flipped vertically, it appears the same.
Letters HIOX can be formed so that they are symmetrical over both a horizontal line and a vertical line. (They can be flipped vertically or horizontally and they appear the same.) Shapes that are unaltered by two flips also have what is called rotational symmetry, that is, a copy will be exactly the same as the original if it is rotated, in these case by 180°. This is sometimes referred to as point symmetry because there is a point at which the rotation must take place so that a copy of the original will map back onto itself.
If formed as a circle, the letter O is symmetrical over any line drawn though its center and has infinite rotational symmetry.
There are three letters of the alphabet that do not have mirror or reflective symmetry but which can be formed so that they have rotational symmetry: NSZ. Notice that the S looks top-heavy. We are accustomed to a larger lower loop, so when the two loops are identical, the letter looks strange.
That leaves letters FGJLPR. Most of them also can be formed with symmetry if one is willing to use lower-case or script versions of the letters. The script L can have rotational symmetry and a script J can have reflective symmetry. (The line of symmetry for the J is offset a bit from horizontal.) A script version of the lower-case l can be formed with vertical symmetry and many san serif typefaces produce the lower-case l as a vertical line, which has both vertical and horizontal symmetry.
The lower-case g shown below has vertical symmetry.
An f with a descender can have rotational symmetry.
A script r with reflective symmetry looks best if the line of symmetry is offset a bit from vertical.
That leaves the letter P. The best I could come up with is to form it as the letter thorn, which is still used in Icelandic.
Below is a symmetrical alphabet. I can think of no use for it.
Numbers are harder to form symmetrically. Here is one attempt.
The 4, 7, 6, and 9 have mirror symmetry over a diagonal line. There may no good way to form 6, 7, & 9 with symmetry.
Eight can have both horizontal and vertical symmetry, but the result is less appealing than one with only vertical symmetry.