When we were young they would break our hearts 
You and I had to go undercover
We knew in time we would be pulled apart
We found a way we could hold on to each other 
When not in physical reality, entities ('souls', if you will, or 'personalities') recognize each other and may incarnate to the physical with the same entities again and again, albeit in different genders, relationships, races, etc. The theory of 'soul mates' is possible here, "when we were young" carrying the possible meaning of "when we were less evolved" or, "before we were born here".

"They" might well refer to guides, teachers, or those who would counsel less evolved entities toward experiences that would further their understanding of reality. "Break our hearts" could possible indicate plans to reincarnate as personalities who would have a negative relationship. "Undercover" may indicate the reincarnated state in which the consciousness does not remember non-physical existence. "In time" could, in this interpretation, refer to physical reality's linear time which does not exist in the non-physical and, therefore, metaphorically refer to physical reality itself.

"Pulled apart" could indicate the knowledge that plans for the physical consciousnesses of the two entities would not include a close relationship and that this is inevitable for the sake of development. One way entities could 'hold on to each other,' is to include some sort of agreement in those plans that would assure reincarnating in the same time period and location. "We found a way " indicates some plan but gives no details.

Gave our souls to another world
Sealed in blood with a sacred vow
Now we've forgotten but beneath all this 
A part of you is still there now
It is, indeed, like 'another world' when the physical consciousness cannot remember the freedom and timelessness of non-physical reality. When one reincarnates, one necessarily agrees to follow the rules of physical reality--rules of which entities are acutely aware prior to entering a body. These rules include laws of physics, perceiving only four dimensions, maintaining a body, birth, death, etc. "Sacred vow" could indicate this agreement by the soul and 'sealed in blood' could mean, in this context, the higher self bound by the agreement to a flesh and blood body. The speaker seems to recognize the higher self of the other entity. ***
go on and sing, go on and dance
it'll all come back if you just give it a chance
Don't you remember me?
We had a secret society
Refresh your memory
We were the secret society
Even when one carries out the "song and dance" plans made for physical existence, it is possible, through meditation, training and practice (or the use of certain drugs, if one is prepared properly or is sufficiently lucky, or sufficiently evolved) to be aware of the true nature of physical reality. Then an entity may become aware of h/his goals in the flesh, and might recognize those with whom those plans would be carried out, for example. The speaker refers, in this context, to the plan the two made for the current incarnation.
I never see you in the waking world
We make an effort to avoid each other 
But every night we penetrate the veil
Where everybody acts like sister and brother 
According to some metaphysical theories, entities inhabiting bodies in physical reality leave that body during sleep periods (what some call astral projection) to continue activities towards spiritual evolution and/or to meet with other entities with whom they have made plans. The physical consciousness usually does not remember these activities, but can be trained to recognize dreams the physical brain makes of them.

During these 'projections' entities are much more like they really are - that is, they can remember their origins and non-physical relationships they have with other entities if they so desire. Although still connected to the physical body, they can react as they had prior to reincarnation. Thus, "Where everybody acts like sister and brother" could indicate the knowledge in the sleep state of a soul's true nature--love and emotions are freely felt, 'love' in its purest sense--genderless benevolence for each other. Utopia?

The lines "I never see you in the waking world/We make an effort to avoid each other" give insight to the overall plan in which the two entities are involved in the physical. They are seemingly at odds and may even be enemies. In the sleep state, however, they would meet often, and know the true closeness they actually share. So, the last lines:

You never speak the words in broad daylight 
They're much too delicate and too naive
would indicate further the physical relationship. "Too delicate and too naive" could refer to that non-physical, genderless reaction to each other, which would, indeed, seem inappropriate and naive to onlookers in physical reality who see the speaker and the one to whom he speaks, especially if they are the same sex and are seemingly at odds with each other.
But everybody's been a witness sometime 
It doesn't change it if you don't believe
These last lines have several possible metaphysical meanings. One could be that the speaker is saying everyone has had experiences that indicate true reality and reality exists whether or not one believes in it. Another interpretation could be that everyone in the physical has seen efforts to improve negative relationships work out successfully, and if one does not believe it possible to accomplish, the negative relationship can not change.

Overall, the song implies, simply, 'We all come from the same place and there's a reason we are in this situation together, even if it is not obvious. Let's not let negative feelings get in the way of our goal.'

Kate Erickson