Edit in JSFiddle

var cube = document.querySelector('.cube');
var radioGroup = document.querySelector('.radio-group');
var currentClass = '';
var showClass = '';

function changeSide() {
  var checkedRadio = radioGroup.querySelectorAll('input');
 
  while(currentClass == showClass || showClass == undefined) {
  var randomValue = Math.floor(Math.random() * 5);
  showClass = 'show-' + checkedRadio[randomValue].value;
  }
  
  if ( currentClass ) {
    cube.classList.remove( currentClass );
  }
  cube.classList.add( showClass );
  currentClass = showClass;
}
// set initial side
setInterval(changeSide, 500);
/* changeSide(); */

radioGroup.addEventListener( 'change', changeSide );

<div class="scene">
  <div class="cube">
    <div class="cube__face cube__face--front">1</div>
    <div class="cube__face cube__face--back">2</div>
    <div class="cube__face cube__face--right">3</div>
    <div class="cube__face cube__face--left">4</div>
    <div class="cube__face cube__face--top">5</div>
    <div class="cube__face cube__face--bottom">6</div>
  </div>
</div>
<p class="radio-group">
  <label>
    <input type="radio" name="rotate-cube-side" value="front" checked /> front
  </label>
  <label>
    <input type="radio" name="rotate-cube-side" value="right" /> right
  </label>
  <label>
    <input type="radio" name="rotate-cube-side" value="back" /> back
  </label>
  <label>
    <input type="radio" name="rotate-cube-side" value="left" /> left
  </label>
  <label>
    <input type="radio" name="rotate-cube-side" value="top" /> top
  </label>
  <label>
    <input type="radio" name="rotate-cube-side" value="bottom" /> bottom
  </label>
</p>

@keyframes dice {
  0% {
    transform: translateZ(-100px) rotateX(0deg) rotateY(0deg);
  }
  50% {
    transform: translateZ(-100px) rotateX(180deg) rotateY(180deg);
  }
  100% {
    transform: translateZ(-100px) rotateX(360deg) rotateY(360deg);
  }
}

* { box-sizing: border-box; }

body { font-family: sans-serif; }

.scene {
  width: 200px;
  height: 200px;
  border: 1px solid #CCC;
  margin: 80px;
  perspective: 400px;
}

.cube {
  width: 200px;
  height: 200px;
  position: relative;
  transform-style: preserve-3d;
  transform: translateZ(-100px);
  transition: transform .5s cubic-bezier(0.025,0.025,0.025,0.9);
  /* animation: dice 2s infinite linear; */
  
}

.cube.show-front  { transform: translateZ(-100px) rotateY(   0deg); }
.cube.show-right  { transform: translateZ(-100px) rotateY( -90deg); }
.cube.show-back   { transform: translateZ(-100px) rotateY(-180deg); }
.cube.show-left   { transform: translateZ(-100px) rotateY(  90deg); }
.cube.show-top    { transform: translateZ(-100px) rotateX( -90deg); }
.cube.show-bottom { transform: translateZ(-100px) rotateX(  90deg); }

.radio-group {
  display:none;
}

.cube__face {
  position: absolute;
  width: 200px;
  height: 200px;
  border: 2px solid black;
  line-height: 200px;
  font-size: 40px;
  font-weight: bold;
  color: white;
  text-align: center;
}

.cube__face--front  { background: hsla(  0, 100%, 50%, 0.7); }
.cube__face--right  { background: hsla( 60, 100%, 50%, 0.7); }
.cube__face--back   { background: hsla(120, 100%, 50%, 0.7); }
.cube__face--left   { background: hsla(180, 100%, 50%, 0.7); }
.cube__face--top    { background: hsla(240, 100%, 50%, 0.7); }
.cube__face--bottom { background: hsla(300, 100%, 50%, 0.7); }

.cube__face--front  { transform: rotateY(  0deg) translateZ(100px); }
.cube__face--right  { transform: rotateY( 90deg) translateZ(100px); }
.cube__face--back   { transform: rotateY(180deg) translateZ(100px); }
.cube__face--left   { transform: rotateY(-90deg) translateZ(100px); }
.cube__face--top    { transform: rotateX( 90deg) translateZ(100px); }
.cube__face--bottom { transform: rotateX(-90deg) translateZ(100px); }

label { margin-right: 10px; }