I haven't figured it out per se, but here's a really good explanation from the blog.

https://elsaelsa.com/astrology/2010/06/07/demystifying-astrology-what-about-aspects-out-of-sign/

I can think of an example for me, maybe.  Chart ruler Mercury buried deeply in the 12th house in Taurus.  Conjunct Ascendant and sun in Gemini, and Saturn rounds out the string.

I am deliberate (Taurus!) in my speech and communication, but it's still FAST (Gemini.)  I think fast.  I talk fast.  I type fast.  I lay the groundwork (vocabulary, knowledge, et cetera) so that when I have thoughts to share, they come out when and how I want them to.  Because I have the foundation to build upon and I know where everything is stored, it's both fast and thoughtful.

The way that worked for me (this is funny to me, given my cancer stellium) but Elsa explained it to me once like this.  Think of the zodiac like a house, you're in one room and their are in another.  If you have out of sign aspects,  like an out of sign conjunction, you are in connecting rooms.  You're focus would be say for the kitchen (cancer) while they are in the den (lots of media or books, think Gemini) so you can see each other, but your focus would still be different yet have a connection to each other.  You might have to work a little harder to communicate (yell from room to room).  I hope that makes sense, it was brilliant the way it was explained to me
Angie 

MagicZara, I think it is important. Have Saturn in the late degrees of Virgo in a position to make a lot of out of sign aspects. Conjuncts Moon and Mercury in Libra, sextiles Venus in Leo, trines MC in Sagittarius and Asc. in Aquarius.

I like the explanation of the conjunction with being in different rooms yelling to each other. Sometimes they don't hear each other right, that's when Moon-Mercury has a big mouth like Muhammad Ali. When they do hear each other, that's when I'm all cautious of what I'm saying and displays of emotions are down. I think there is a middle ground to this. Virgo is a rational sign and Saturn is a teacher. Moon-Mercury mixes mind (thoughts) and heart (feelings) and with Libra, you can be undecisive. Saturn helps Mercury and Moon to get thoughts and feelings sorted out, to be able to make a logical decision.

Your Virgo-Scorpio square is tricky. I wonder if early degree planets still have some qualities from the sign they left and if the late degree planets apply the qualities of the sign they are about to enter - and this gives some interference. (?????)

Bumping this because last night, out of nowhere, this issue entered my mind. I guess it was because of the discussion of the 29th degree. Having 3 planets in the 29th degree, I've had cause to think about out-of-sign-aspects in my natal chart, and in transits and synastry. Am I off the hook when Saturn goes to 0 Sag or 0 Virgo, or still in the midst of a close opposition or square to my Sun?

It seems to me that the "debate" about out-of-sign aspects has a lot to do with how you conceive of aspects in the first place. Many years ago, when I was first getting interested in astrology, someone gave me The New Astrologer by British astrologer and literary critic Martin Seymour-Smith. Seymour-Smith was big on harmonics, and viewed aspects primarily in terms of wavelengths. A square, at 90 degrees, is 1/4 of the circle so it takes 4 90-degree wavelengths to add up to 360 degrees. Under this theory the degrees of separation are all that matter. Seymour-Smith defines "dissociate," or out-of-sign, aspects in the book but offers no real interpretation.

The other theory of aspects is that the nature of the aspect (squares are "hard" and trines are "easy) comes from the relationship between the affected signs. This is how most people learn aspects initially. Thus a trine from 0 Aries to 2 or 3 Leo makes sense. A "trine" from 0 Aries to 29 Cancer, though closer in orb, doesn't make sense. Aries and Cancer are in tension and usually form a square, not a trine. Likewise, a square from 29 Aries to 29 Cancer makes sense, but one from 29 Aries to 0 Leo does not. For this reason some astrologers dismiss out-of-sign aspects out of hand.

What came into my mind is the sesquiquadrate, the 135-degree aspect. Seymour-Smith believed this "8H" aspect (based on a multiple of 45 degrees, or 1/8 of the zodiac) almost as strong as a square. It's interesting because any planet will make a 135-degree aspect with a point in a supposedly compatible sign of the same element. For example, a planet at 5 Aries makes a 135-degree angle with 20 Leo and with 20 Scorpio. A tough aspect between Aries and Scorpio, which "normally" quincunx, makes more intuitive sense than a tough aspect between Aries and Leo. I wonder if it turns out that way or if the 135 degrees are all that matter.

My own sense is that an aspect that's close to exact by degree is very real, but the signs will have an muting impact. I have felt some relief when a difficult planet crosses into the next sign, even if it's still within half a degree of aspecting my Sun or Mars.

Anyone had any experiences with this kind of thing?

Thanks for bumping, S and T. Nice analysis. I have by no means figured this all out, but I've come to the conclusion, based on my own life, that if you have an out-of-sign hard aspect involving two "friendly" signs, the hard aspect is easier to cope with. Thus, yes, there is a muting that happens. Case-in-point: My Moon-Uranus square means my home life is rarely stable, but I'm mostly able to roll with it regardless. I have much more trouble with my Mercury-Neptune square, which is a "proper" in-sign square.