Physics with intact protons at the LHC: from the odderon discovery to the sensitivity to beyond standard model physics

We describe the discovery of the colorless $C$-odd gluonic compound, the odderon, by the D0 and TOTEM Collaborations by comparing elastic differential cross sections measured in $pp$ and $p\bar{p}$ interactions at high energies. We also discuss the reach on quartic anomalous couplings and the sensitivity to axion like particle production by using the LHC as a $\gamma \gamma$ collider and detecting the intact protons at high luminosity.