Numbers with Rotational Symmetry

Python 2, 50 bytes

lambda n:`n`==`map('01xxxx9x86'.find,`n`)`[-2::-3]

The method '01xxxx9x86'.find takes a digit character to it's upside-down number, with any unflippable digit giving -1. This function is mapped to the reversed number string, producing a list of digits.

This is converted to a string with the [1::3] trick, except it's reversed instead by doing [-2::-3] (thanks to Dennis for this, saving 4 bytes), and compared to the original number string. Any -1's from unflippable digits will misalign the conversion, making it fail.

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56 bytes:

lambda n:`n`[::-1]==`n`.translate('01xxxx9x86______'*16)

Checks if the number string reversed is the same as it with the upside-down replacements. Digits that can't be flipped are replaced with 'x' to always give the wrong answer.

The replacement is done with translate on a string of 256 chars, replacing the corresponding ASCII values. Only the 10 values 48 to 57 matter, but I padded to length 16 to make the total length be 256. I wonder if there's a shorter way.

Some other approaches (lengths 59, 60, 60):

lambda n:set(zip(`n`,`n`[::-1]))<=set(zip('01896','01869'))
r=lambda s:set(zip(s,s[::-1]));lambda n:r(`n`)<=r('9018106')
lambda n:all(a+b in'001188969'for a,b in zip(`n`,`n`[::-1]))

05AB1E, 22 16 15 14 bytes

Code:

Â23457ð«-69‡Q

Try it online!

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Previous code:

Â69‡Q¹¹„vd•ÃQ*

To figure out if the string is rotational symmetric, we just need to transliterate 69 with 96, reverse the string and check if they are equal. The other thing we need to know is if the number only contains the digits 0, 1, 8, 6 and 9. So that's exactly what we are going to do:

                     # Bifurcate the input, which pushes input and input[::-1]
 69Â                  # Bifurcate 69, which pushes 69 and 96.
    ‡                 # Transliterate 69 with 96 in the input, e.g. 1299 becomes 1266.
     Q                # Check for equality.
      ¹¹              # Push input again twice.
        „vd•          # Compressed version for 10869.

The „vd• part actually converts the string vd from base 190 to base 10. You can try this out here.

            Ã        # Keep the characters of the second string in the first string.
                       667788 would become 6688 (since 7 is not in 10869).
             Q       # Check for equality.
              *      # Multiply the top two numbers, which actually does an AND function.

Uses CP-1252 encoding. Try it online!

Ruby, 54 46 bytes

 ->a{(b=a.to_s).tr('1-9','1w-z9x86').reverse==b}

I don't know, is anonymous functions like that allowed or not

Basically same idea as Python2 answer. If input is not integer, act bad (i.e. aba gives true)