If your average gentile passed by a group of Jews saying קדוש לבנה they would probably be quite confused - not because a group of individuals appear to be blessing the moon but because they see ten Jews facing eleven directions. Many people seem to face all sorts of different directions in order not to face the moon. While this seems like a perfectly innocent practice, there are some serious issues involved.
The reason why people do this is so as to not appear to be gazing at the moon (either so as not appear to be praying to or worshiping the moon or simply to avoid the prohibition to gaze at the moon.) This is certainly a concern that is brought up in halachah. Mishnah Berurah (426:13-14) quotes from poskim that one should not gaze at the moon, even during the berachah. Rather, one should give a quick glance beforehand. He also states that we dance by standing on our toes because the bending of our knees might appear as bowing. However, nowhere, does it mention that one should turn away from the moon. I do not know of any other sources who do suggest to turn away but what I do know is this: It is nothing short of blasphemy to direct the following words at HaShem:
ברוך יוצרך, ברוך עושך, ברוך קונך, ברוך בוראך
These words must be directed at the moon. I would think that the same would hold true for כשם שאני רוקד כנגדך ... which again is certainly talking about the moon. I am not sure if people just don't know what they're saying or just aren't thinking but I myself have seen people face away from the moon towards the shul from the beginning of קדוש לבנה until the end. I think this an unfortunate example of over-zealousness gone awry. I might even go as far as to put this in the same league as people turning their backs on ברכת כהנים - a practice which is fortunately diminishing but is still found here and there. I feel rabbanim should bring this to their congregants' attention.
UPDATE: A reader has brought to my attention two sources which actually do discuss this practice.
I nevertheless still maintain that the source for this is obscure, at best. And certainly of note is the point made by יסוד ושורש העבודה to which I was מכוין, regarding those two phrases. If there is a concern related to gazing at the moon, I still do not feel that is reason to turn away. But at the very least, if that is to be the practice, everyone ought to get together and decide to face the same direction. In או"ח צ"ד, it is made very clear how careful one must be with the direction they face for שמונה עשרה, so as not to make the ציבור appear disjointed. While this is outside of the בית הכנסת, I think the same care should be taken. (Especially now considering the יסוד ושורש העבודה suggesting that one needs to stand in the same manner as in שמונה עשרה.)
UPDATE: A reader has brought to my attention two sources which actually do discuss this practice.
:לבושי מרדכי
:יסוד ושורש העבודה
I nevertheless still maintain that the source for this is obscure, at best. And certainly of note is the point made by יסוד ושורש העבודה to which I was מכוין, regarding those two phrases. If there is a concern related to gazing at the moon, I still do not feel that is reason to turn away. But at the very least, if that is to be the practice, everyone ought to get together and decide to face the same direction. In או"ח צ"ד, it is made very clear how careful one must be with the direction they face for שמונה עשרה, so as not to make the ציבור appear disjointed. While this is outside of the בית הכנסת, I think the same care should be taken. (Especially now considering the יסוד ושורש העבודה suggesting that one needs to stand in the same manner as in שמונה עשרה.)