You can't be really be truthful until you are at least truthful to yourself, at least according to my understanding. Because if we actually know the mind, it is full of its justifications, isn't it? But both are very important, being truthful to ourselves and being truthful to others. But I will put a caution about truthfulness to others, truthfulness to others has to be balanced with loving-kindness, because we don't have to tell everybody what's in our mind. And we have to be careful about whether our telling the truth to others is based in good intention or unbeneficial intention. That is why truthfulness to ourselves is most important, to actually see what our intention is for saying something to others. If we are truthful to ourselves then we can see whether our wish to be truthful to others is actually based in the compassionate intention or whether it is based in a selfish intention. We have to see what is appropriate.
Right Speech also deals with appropriateness of speech. Sometimes we have to be very careful and skillful with our speech so that we are both truthful and loving-kind. If the truth is something that is not appropriate to say at that time to someone else, then we learn how to avoid answering in ways that may be unkind to others. Sometimes not speaking is better then actually saying the truth because we are also wishing for a skillful outcome to the situation. So, it is sometimes difficult to know when to say the truth and when to be quiet and not say anything, or to actually just ask a counter-question to avoid difficult situations.
But being truthful to ourselves is very important. Otherwise, it is difficult to come to know what is the truth, because often what we think is the truth, is not the truth, because we are only seeing it from our perspective with our opinions and views about things which may be prejudicial. So, it is important before we actually form a frozen idea of what the truth is, to investigate into it and to see it from many perspectives, and get more information. Truthfulness is a very important Parami because actually all of us, I hope, are trying to understand the truth, the Four Noble Truths. And in order to understand the truth we have to begin with the truth. But most of us, because we have ignorance, have wrong view, so in building the power of wisdom, we come closer to understanding the truth, what is the truth. Mindfulness of our thoughts, and understanding cause and effect, and the Four Noble Truths as it arises in our mind and body help us come closer to knowing what is the truth.