Perl / JavaScript, 74 bytes

Perl returns a palindrome and JavaScript 'de-palindomizes' the input string.

s=prompt();console.log(s.slice(0,s.length/2+.5|0));eg=1;t=eg;s=$=reverse$_=eg

JavaScript

Assumes the string is a valid palindrome already.

All the work happens in the first section:

s=prompt();console.log(s.slice(0,s.length/2+.5|0));

Stores input in s, then returns the first half (rounded-up) of the string. The rest of the script is pointless variable assignments:

eg=1;t=eg;s=$=reverse$_=eg

Try it here:

s=prompt();console.log(s.slice(0,s.length/2+.5|0));eg=1;t=eg;s=$=reverse$_=eg

Perl

Must be run with -pl, eg:

perl -ple 's=prompt();console.log(s.slice(0,s.length/2+.5|0));eg=1;t=eg;s=$=reverse$_=eg' <<< 'test'
# returns testtset

The code is basically two substitutions (s///), but using = as the delimiter instead of /:

s/prompt();console.log(s.slice(0,s.length\/2+.5|0));eg/1;t/eg;s/$/reverse$_/eg

The first, replacing prompt();console.log(s.slice(0,s.length\/2+.5|0));eg (which, admittedly will mess up your palindrome that contains "prompt;console.logs.slice0;eg"...) with 1;t which is evaled, returning t, the second replaces the end of string $ with reverse$_ which is then also evaled and appends the original source string reversed.

Try it online!