Rotate 45-degree element around vertical axis

The trick is to set this rotation before the 45 degrees rotation:

Notice also that to make the rotation behave really as expect, you need to set it to 0 in the base state

.container {
    width: 200px;
    height: 200px;
    margin: 100px;
    border: solid 1px;
    transform: rotateY(0deg) rotate(45deg); /* needs Y at 0 deg to behave properly*/
    transition: transform 2s;
}


.container:hover {
    transform:  rotateY(180deg) rotate(45deg); /* notice the order */
}
.inner {
    margin: 50px;
    transform: rotate(-45deg);
}

<div class="container">
<div class="inner">INNER</div>
</div>

This is how I interpret the question. I'm not very happy with the demo since it needs a lot of structure. But maybe you can verify the behavior?

Basically I use a wrapper to rotate on the y-axis. It is key to set the transform origin to the center. The additional wrapper is used to prevent a flickering on mouse hover.

https://jsfiddle.net/nm59mqky/1/

.tile {
  transform: rotateY(0);
  transform-origin: center center;  
}

.wrapper:hover .tile {
  transform: rotateY(180deg);
}