How does CNN deal with rotation invariant pictures?
With a standard CNN, features are not rotation invariant, and the are not rotation equivariant.  They are translation equivariant, but not rotation equivariant.
If you would like the model to be rotation invariance, then there are fancy methods to provide that, and simple methods.
The fancy method is to use a rotation-equivariant neural network.  There are many research papers on this subject.  See, e.g., Deep Rotation Equivariant Network by Li et al., Learning Steerable Filters for Rotation Equivariant CNNs by Weiler et al., and many others.
The simple method is to use data augmentation.  During training time, randomly rotate the input image before feeding it to the neural network.  You might need to train the neural network for more epochs.
Data augmentation is a lot easier to implement, and my impression is that it gives results that are close to what can be achieved with the fancy methods.  Therefore, I suggest you try data augmentation first.