What can open up the cuticle? Commonly, chemical processing of hair such as relaxing or colour rely on alkaline products and these CAN open up the cuticle. Your average shampoo does not have the ability to do this. They are mostly mildly acidic to neutral (around pH 6-7 though 8 is considered ok) because cosmetic scientists need to consider that the shampoo can end up in your eyes, face and mouth, so it is safer to be close to neutral. See this site for some pH ranges of shampoo. Lisa Akbari also advertises a shampoo at pH 5. The pH of hair and skin is a range between 4.5 to 5.5.
Therefore, if your hair is natural, you most likely have a closed cuticle as has been seen scientifically (Khumalo et al, Journal of the American Academy of Dermatology, 2000, pp 814-820).
This leads to the question, if the cuticle is closed then what can really get in? Well two things can, water and some oils.What kind of oil can get into the hair cortex? The complex answer is many different oils to varying degrees. The simple answer is coconut oil and olive oil have been shown to penetrate hair quite well (Hornby et al, Int Journal of cosmetic science 2005 pg299-300).
This now leaves the question, what exactly is the purpose of conditioner?Well most conditioners do just work on the outside. They soften the hair, help smooth the cuticle, minimize static, sometimes help the water stay in the cortex. When rinsed off they do leave some conditioning agents behind which keep the hair feeling soft.