Let's take Libra, first. Libra is a masculine sign, as it's ruled by the Air element. Air is one of the two masculine elements in astrology (along with Fire). Air is symbolic of logic, objective thought, and solid reasoning. This sort of rationality is the way in which males are conditioned to behave in our society, valuing reasoning over emotion. This is usually the manner in which Libra-influenced people conduct themselves, believing that it's always better to think things out than get carried away with raw feeling.
However, Libra is also ruled by Venus, which is one of the feminine planets. Venus stands for beauty, appearance, pleasure, and sensuality; all rather feminine attributes. Taurus is also ruled by Venus, but is an introvert sign, meaning that it's pleasure and sense of harmony is directed inward and toward itself. Libra, being extrovert, directs that pleasure outward. This results in a people-pleasing manner that Libra is notorious for which reflects our society's female values: that everyone should be happy and everything should be pleasant and that the boat should not be rocked.
Libra's masculine-feminine energy is combined in the manner that they conduct themselves rationally, yet can be indirect and downright passive about their own needs. They are more comfortable focusing on other people's needs and don't like too much aggression. Men strong in Libra energy are usually very gentle, refined, and soft, unlike the traditional male that society idealizes. It's with the males that the androgyny of this sign makes itself most well-known, as they are bound to not exactly fulfill the ultra-aggressive, assertive ideal that men are expected to live up to.
Libra-influenced women can be quite a different story, though. They can usually come off as perfectly charming and dainty little ladies. With Libra, the more feminine energy is what is most obvious, as Venus rules it, on the surface. Therefore, Libra females tend to fit more easily into society's feminine ideal. However, they will be the types that lead much more with their head than their hearts. They are highly objective women, valuing grace and reason over uncontrolled emotion.
Now, let's talk about Scorpio, whose androgyny manifests in the complete opposite manner. Scorpio is a feminine sign, one of the three in the Water triplicity. Water, along with Earth, is feminine. The Water element represents depth, intuition, reflection, and emotion. These are all more yin-based values for obvious reasons. The depths that women are more actively encouraged to express and venture down into it are where Scorpio-influenced people, both male and female, feel most at home.
Yet, Scorpio's traditional planetary rules is Mars. This is a masculine planet. Mars, also the ruler of Aries, symbolizes assertion, aggression, drive, and the courage to face fears and challenges. Aries is an extrovert sign, so all of this strength is directed outward into the world. But, introverted Scorpio directs the power of Mars deep into their soul. This results in the formidable sort of inner strength that people with a prominent Scorpio influences naturally exude. Regardless of gender, they have a male-like quality that says they're not to be messed with. They seem to not be afraid (Mars) of just about any emotional territory (Water), no matter how much it might scare others.
While Libra is feminine on the outside but driven by a masculine energy, Scorpio is the opposite. Mars makes Scorpio-influenced individuals more masculine on the outside, but motivated by feminine concerns. They care a lot about their emotions, yet they will do their best to preserve them. Scorpio is way more sensitive than people know, as the unshakable strength they exude often hides or obscures their depths of feeling. It can be very tough for them to let others see them cry, even if they do quite often. Scorpio females are most apt to defy society's expectations of females. They will be strong, efficient, and resourceful, with an active need to find and own their power in life. As deeply emotional as she is, she will never be a helpless damsel in distress and will actually use those feelings to focus and drive her.
The strength that these women exude can make them intimidating and often easily maligned in our society. Scorpio-influenced men tend to have an easier time of it, as the sign's masculine values of power, strength, and owning one's sexuality are more encouraged in men than women. Still, do not be fooled by this male's tough exterior, as there is going to be a lot going on inside emotionally. The extraordinary instincts of their sign give them the kind of intuitive hunches and insight that are quite feminine in nature. He is soulful and perceptive and this depth only enhances this inner strength and confidence in the end.
It's easy to see why these signs can be at such loggerheads. As androgynous as they both are, it's in very different ways. Libra's feminine need to please and pacify will ultimately serve their goal, as a masculine sign, to find intellectual understanding and connection with other people. Scorpio's masculine fearlessness and strength is ultimately what allows this feminine sign to access an uncommon level of raw emotional power and intensity, as well as a deep connection with and insight into those around them.