The camera would need to be quite a bit larger and heavier in addition to being more complex and fragile.

We've been there before, back in the film days, when the popular medium format cameras were 6x6 square format SLRs like the Hasselblad 503cx and the Rollei 6006, and Mamiya's rotating-back 6x7 format RB67 and RZ67. Because the Mamiya backs rotated, they actually had to be 7x7 cameras (the 6cm dimension was the width of the film roll). That meant that the mirror and focusing screen had to be big enough to support an image that was 7cm in either direction. The Mamiyas were huge compared to the 'Blads, and that was just to get a 4:5 aspect ratio working in both orientations.¹ It would be considerably worse with a 2:3 aspect ratio.

Because an APS-C-sized sensor is as wide as a full-frame sensor is tall, you could fit the works into a standard SLR mirror box without modifying the lens system, but to make it work with a full-frame DSLR, you would need to use a different lens mounting system with a longer register distance to accommodate the larger mirror. You would also need to worry about masking the viewfider for the format currently in use, and creating a viewfinder that covers the larger screen with a reasonable eyepoint and magnification (and VF info somewhere near the image without occulting anything).

In addition to building the reflex mirror and viewing screen big enough to support both horizontal and vertical images, a focal-plane shutter would need to travel the long dimension of the sensor as well. An APS-C-sized sensor would need, in essence, a full-frame shutter, and a full-frame camera would need the shutter from a Leica S. Either the sync speed is going to come way down, or the cost is going way up.

Mirrorless cameras would make this scenario a lot more practical, at the expense of making the body itself slightly bigger and heavier. And, of course, more mechanically complex and fragile. The lack of mirror and focusing screen, at least, means that the lens mount would not need to change.

Lens hoods, though, would need to be less effective. The current "tulip" shape in use is designed to closely fit the rectangular image without vignetting. If the sensor rotates, the lens hood would either have to be round or square to accommodate both formats.

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¹ "6x6" was actually 56mm x 56mm, and "6x7" was 56mm x 70mm. The film was 6cm wide, but the image had a 2mm border.