The world is like a Mirror: smile and it'll sɯıןǝ qɐɔʞ

Pyth - 174 bytes

Can prolly save with base compression or something cuz of unicode(this is only 119 chars)

u?qH\_&=hZG+G@@c2K+s+rB;1UT".?! ɐqɔpǝɟɓɥıɾʞlɯuodbɹsʇnʌʍxʎz∀ƆᗡƎℲ⅁HIſ⋊⅂WNOԀΌᴚS⊥∩ΛMX⅄Z0⇂ᘔƐ߈ϛ9ㄥ86˙¿¡ "ZxKH?}\|zs_Bhcz\|zk

Test Suite.

JavaScript (ES6), 308 bytes

s=>s[r='replace'](/_.*?(_|$)/g,m=>m[r](/./g,c=>'| ɐqɔpǝɟɓɥıɾʞlɯuodbɹsʇnʌʍxʎz∀ƆᗡƎℲ⅁HIſ⋊⅂WNOԀΌᴚS⊥∩ΛMX⅄Z0⇂ᘔƐ߈ϛ9ㄥ86˙¿¡'['| abcdefghijklmnopqrstuvwxyzACDEFGHIJKLMNOPQRSTUVWXYZ0123456789.?!B'.indexOf(c)]||''))[r](/(.*)\|.*/,(m,t)=>t+[...t].reverse().join``)

Test

var solution =

s=>
  s[r='replace'](/_.*?(_|$)/g,m=>
    m[r](/./g,c=>
      '| ɐqɔpǝɟɓɥıɾʞlɯuodbɹsʇnʌʍxʎz∀ƆᗡƎℲ⅁HIſ⋊⅂WNOԀΌᴚS⊥∩ΛMX⅄Z0⇂ᘔƐ߈ϛ9ㄥ86˙¿¡'
        ['| abcdefghijklmnopqrstuvwxyzACDEFGHIJKLMNOPQRSTUVWXYZ0123456789.?!B'
          .indexOf(c)]||''
    )
  )
  [r](/(.*)\|.*/,(m,t)=>t+[...t].reverse().join``)

var testCases = [
  '_Will I be mirrored?!',
  'Wi_ll I be mi_rrored?!',
  'Will I be mirror|ed?!',
  'This_ is a test_ cont_aining bo|t_h mirrors.',
  '0_1_2_3_4_5|_6_7_8_9'
];
tests.textContent = testCases.map((c) => c + '   =>   ' + solution(c)).join('\n');

<input type="text" oninput="result.textContent=solution(this.value)" value="This_ is a test_ cont_aining bo|t_h mirrors." /><pre id="result"></pre><pre id="tests"></pre>

PERL 243

242 + 1 for -p

$h=(split'\|',$_)[0];$_=$h.reverse$h if($_=~/\|/);for$b(split'_',$_){$b=~y/A-Za-z0-9.?!/ɐqɔpǝɟɓɥıɾʞlɯuodbɹsʇnʌʍxʎz∀ƆᗡƎℲ⅁HIſ⋊⅂WNOԀΌᴚS⊥∩ΛMX⅄Z0⇂ᘔƐ߈ϛ9ㄥ86˙¿¡/ if($f);$\.=$b;$f=!$f;}$_='';

Ungolfed:

$reverse = (split('\|', $_))[0];
$_ = $reverse . reverse($reverse) if($_=~/\|/);

for $block (split '_', $_) {
    $block =~ y/A-Za-z0-9.?!/ɐqɔpǝɟɓɥıɾʞlɯuodbɹsʇnʌʍxʎz∀ƆᗡƎℲ⅁HIſ⋊⅂WNOԀΌᴚS⊥∩ΛMX⅄Z0⇂ᘔƐ߈ϛ9ㄥ86˙¿¡/ if($flip);
    $\.=$block;
    $flip=!$flip;
}
$_='';

Example:

$ perl -p mirror.pl <<<'Will I be mirror|ed?!'
Will I be mirrorrorrim eb I lliW

I'm 99% sure this works, but most of the upsidedown characters don't display correctly on my system. I tested it by using a different range of character, e.g. .?! -> %^& instead of .?! -> ˙¿¡.