The core for me is respect and a spirit of research. For example my major influence has been Inaba sensei of the Meiji Jingu Shiseikan - he frequently talks about and demonstrates how his own Shinto beliefs are the core of his Budo. But he also says that he is not seeking to convert people, he is instead challenging them to find their own core consistent with their culture and heritage. There are times when I "channel my inner Inaba sensei" on the mat - I am sure that can look strange to others too.
Here is an interview with Inaba Sensei from the Japan Times website;
"Losing is always a sore subject. Japan lost the war in 1945 and it's still bleeding from its wounds. We have not recovered from the aftereffects of the Occupation. Healing takes time because the wounds are picked again and again so the scabs can never fall off.
Believe that the end is always good. In Shinto, our creation story begins with great happiness and lovemaking and Amaterasu the sun goddess giving birth to many children. Shinto is all about positive feelings and I think this is why Japanese people are optimists and never give up. Even after the atomic bombings in Hiroshima and Nagasaki, we were immediately rebuilding and never blamed the United States ever. Life is just too fun to waste.
We lost the war, but at least we have the Emperor. Japanese culture originates with him. Our nation's foundation is Shinto, with the Emperor as the highest priest who listens to the sun goddess and offers her festivals and ceremonies. He is the one person who knows the feelings and wishes of both the goddess and the people.
In the Japanese language words and emotions are perfectly matched. When foreigners speak Japanese, they are much kinder and gentler than when they use their mother tongue. Their facial expressions soften and they are unable to say yes or no clearly. Japanese food also contributes to this metamorphosis — after five days living here on miso soup and rice, the color of their faces is healthier and they look better."
Now, I find these statements... uncomfortable, but the question is, if your sensei starts publicly making statements like this, what does one do?