I don't know that there is a rule per se but it is important to me that it be pleasing from outside, looking in. Nothing worse to me than to observe a window with a light colored window treatment, next to a red (or whatever) color right next to it. But I consider this a matter of personal taste, not written in blood anywhere.

all my windows that are on east side of house are the same, white shutters or white wood blinds.

All on the opposite end of house are natural wood blinds or shutters.

Rules are for fools, do what you like.

I think it's nice to have a somewhat consistent face on nearby windows but they don't have to be identical. I like things to look essentially "white" on the outside (or neutralish). But I don't think the treatments have to be identical, or even similar.

One thing about bamboo blinds is depending upon the width of the grasses used, they are essentially Transparent when lit from behind. (As in, when it's dark out, the blind is down and the room is lit up). This gives people a false sense of security, I think, because I have had neighbors who would put down their bamboo blinds and then walk around completely naked in front of the windows. So you may want to have at least a sheer lining on the back of the bamboo for added privacy and a more neutral appearance.

It depends... I know - no help at all, but that is the only answer I can give. Ones eye knows if something works or not.

I prefer that all window treatments have a white colored lining, so that the appearance from outside is somewhat consistent. The treatments can be different, such as drapes, roman shades, etc., but the linings should all match.