Yes, sastram has injunctions on not cutting or trimming our hair. For men, there are injunctions on how the kudumi / shika should be , on what days they should not shave the rest of the head and body, and so on.
Likewise, for women it is prescribed not to cut or trim hair. So, it is better to follow your gramdmother’s advise. (It is also sastric statement that “when you are unclear on what is to be done and what is not to be done, consult the eldest lady in the family; they will guide appropraitely”. Therefore பாட்டி சொல்லை தட்டாதே! 🙂
Why we shouldn’t cut our hair? I have not heard any explanation on this with specific references to scriptures. However, I have heard that in occult practices the hair is the place where one’s ‘energy’ is stored. If the occult practitioners cut their long hair their powers will be lost. (These are just hearsays; I do not have proof or refences to support this). Interestingly even Samson in the Samson and Delilah story also had his powers stored in the hair!
So, I presume, the hair may be important in some mystical way which are beyond our limited understanding.
In anycase, it is not too difficult to follow this one. In ancient days women were far more busier than now, with a lot of responsibilities upon their head! So, hair maintenance should be easier now than it was for the bygone generations.
And above all, it looks so elegant and cammands respect from onlookers! Look at all the models how they show-off their long tresses!
So, please don’t cut. Later on if you want it back, it can not grow back quickly, and may be not to the same length!
By the way, your question reminds me of a story by O Henry “Gift of the Magi” (pronounced as ‘me-joy’. It is a very nice story; you may like it.