Two-way Palindromic Closure Generator

Pyth, 22 19

hfq_TTsm,+z_d+_dzyz

Try it online.

Explanation

The two-way palindromic closure is either of the form AX or XA, where X is the input string and A is a substring of X. I actually has to be a contiguous substring of X, a prefix for the one form, a suffix for the other form. But I don't care about these defails. A substring (contiguous or not) is all I need in Pyth.

                        Implicit: z = raw_input() // Read a string
                 yz     A list with all substrings (contiguous or not) of z
       m                For each of these substrings d, build
        ,                  a pair of two strings:
         +z_d              ( z + inveres(d) ,
             +_dz            inverse(d) + z )
      s                 Sum (put all created pairs into a list)
 fq_TT                  filter elements T, where inverse(T) == T (Palindrom)
h                          Take the first element

Edit

The old version ordered the the strings after filtering by length .olN.... Just realized, that y returns the substrings ordered by length. So these palindromes are already sorted.

Clip, 40

(sl`f[a=ava}+m[i+v+ixx}Rlxm[i+xvu0ix}Rlx

Example

Documents>java -jar Clip4.jar palindrome.clip
abcb
abcba

Explanation

(sl`                                        .- The shortest                     -.
    f[a=ava}                                .- palindrome                       -.
            +                               .- among the following two sets:    -.
             m[i      }Rlx                  .- For each number up to length(x)  -.
                +                           .- combine                          -.
                 v+ix                       .- the last i chars of x, reversed  -.
                     x                      .- and x.                           -.
                          m[i       }Rlx    .- For each number up to length(x)  -.
                             +              .- combine                          -.
                              x             .- x and                            -.
                               vu0ix        .- the first i chars of x, reversed.-.

CJam, 30 bytes

Was really hoping to see a CJam answer by now.. So here it goes :P

q:Q,{)QW%/(Q+Q@+s}%{,}${_W%=}=

I really hate that {,}$ block in there, but I get an unordered list of possible palindromes due to the generation algorithm I am using.

Code explanation

q:Q,{            }%             "Read the input string, store in Q, take length and";
                                "run the code block that many times";
     )QW%                       "Increment the iteration index and Put reversed Q on stack";
         /                      "Split reversed Q into parts of incremented iteration index";
          (Q+                   "Take out the first part and prepend it to Q";
             Q@+s               "Take the rest of the parts and append them to Q";
                   {,}$         "At this point, we have all possible prepended and appended";
                                "sequences of the input string. Sort them by length";
                       {_W%=}=  "Take the first sequence which is a palindrome";

Try it online here