I gave D-xhird more play time than it deserved, so I can explain about it's tricks...
The ice arena walls are blended against VDP2 layers. Nothing extraordinary here. Since you can do "mode7" on two VDP2 layers, it's possible to fake 2 3D planes using 2D backgrounds. Same for the semi-transparent cloth walls in that chinese stage.
I remember Karen's skirt being transluscent... it's prolly sprite-to-sprite transparency (always 50% blended). The reason it looks different depending on the angle is due to some pixels in the quads being drawn more than once (and thus becoming more and more opaque at each redraw) when the quad distortion increases.
Pixels in the VDP1 buffer that are blended against the VDP2 have a blend ratio property. Sonic R simply increases the ratio for polygons that are further from the camera, until they fade out completly.
Burning Rangers was a special case. It rendered the same scene TWO times, into two different buffers.
First it draws all the non-transluscent stuff normally, with textures, lighting and so on, to the backbuffer.
Then it draws everything - including transluscent stuff - at half resolution, using black polygons for all opaque geometry (walls, the characters) and drawing transluscent objects normally, to an off-screen buffer that is draw as a VDP2 layer on TOP of the VDP1 buffer using the VDP2 additive blend mode. That's why all additive blended objects looked extra pixelated: because they really were.
How it does that exactly is beyond me - I suspect it uses a software renderer to draw directly to the VDP2 bitmap memory, since I suppose the Saturn doesn't store textures in the VDP1 ram, and therefore cannot "render to texture", like the PSX and N64 can.