Generally, it is ideal for people to be respectful at all times to one another. Because the exchange of respect (or lack thereof) inevitably must occur between two or more people/entities, this ideal notion is not always practically achievable in the real world. What I mean by this is that if one person is being disrespectful and you are being completely respectful, it is ok to be somewhat disrespectful back. If someone does not acknowledge the honor you deserve and demand from simply being human, one can exemplify a receptive attitude that matches the level of respect being shown to them. However, this is only true if the degree of disrespect you show to another disrespectful person has an appropriate magnitude, does not go beyond the level of disrespect you are receiving, and is done in an amount that is necessary to accomplish something. If the response you show back is one acted upon haste, pure emotion (most likely this would be anger or some subset of it), a faulty line of reasoning, or an offense/ attack to another's personal attributes, it becomes exaggerated, unnecessary, childish, and irrelevant. This kind of mentality when exercising appropriate disrespect yields unproductive results and is a waste of time in general. Therefore, when acting "respectfully disrespectful", one should always be wary and cautious of the limits that should not ethically be crossed .
Is not getting along with someone an exception to this rule/standard? Is it a reasonable justification to act disrespectfully? Well, it is a common understanding (ironically) between all people that there are different perspectives about different topics and people do feel differently about things. This acknowledgement of perspective and dismissal of the existence of one "right" or biased perspective is a crucial aspect in exercising respect. Therefore, not getting along with others and possessing different viewpoints is a celebratory affair; it expands upon existing knowledge and can remove unjust prejudice(s) from society. One should learn and practice the skill of mutually coexisting with others in a world that does not always come to one shared consensus.
When one witnesses a situation in which people are being disrespectful, one can effectively intervene by gently reminding the other person that they are attempting to promote peace and not instigate any conflict. This can assure the disrespectful person that it is not necessary to put up a defense of disrespectful behavior. In other words, the person can rest assured knowing that if they show a respectful attitude, they will not be met with the opposite attitude. This balances the situation and presents a commonality between two people. Afterwards, when it is a more settled and calm environment, one can make valid points and express the problems they found in how the other person was treating them. The expected, ideal outcome of this series of actions would be a simple attitude of forgiveness and a shift in attitude.
Personally, respect is an integral part of my character because my parents instilled its importance within me and have set consequences for not being respectful. This is a moral core value I have held within me since I was young and have spread to other people as well. More than just contributing to my ethical standards, showing respect towards others is a means for me to make meaningful and healthy relationships that contribute to overall happiness and standard of living. It is also, I believe, a very underrated and implicit solution to many issues and can prevent conflicts from arising.