  Higher-order spatial laser modes have recently been investigated as
candidates for reducing test-mass thermal noise in ground-based
gravitational-wave detectors such as advanced LIGO. In particular, higher-order
Hermite-Gauss (HG) modes have gained attention within the community for their
more robust behaviors against random test-mass surface deformations and
stronger sensing and control capacities. In this letter we offer experimental
investigations on various aspects of HG mode interferometry. We have generated
purified HG modes up to the 12-th order $\mathrm{HG}_{6,6}$ mode, with a power
conversion efficiency of 38.8% and 27.7% for the $\mathrm{HG}_{3,3}$ and
$\mathrm{HG}_{6,6}$ modes respectively. We demonstrated for the first time the
misalignment and mode mismatch-induced power coupling loss measurements for HG
modes up to the $\mathrm{HG}_{6,6}$. We report an excellent agreement with the
extended numerical power loss factors that in the ``small power loss'' region
converge to $2n+1$ or $n^2+n+1$ for a misaligned or mode mismatched
$\mathrm{HG}_{n,n}$ mode. We also demonstrated the wavefront sensing (WFS)
signal measurement for HG modes up to the $\mathrm{HG}_{6,6}$. The measurement
result is accurately in accordance with theoretical WFS gain
$\beta_{n,n-1}\sqrt{n} + \beta_{n,n+1}\sqrt{n+1}$ for an $\mathrm{HG}_{n,n}$
mode, with $\beta_{n,n-1}$ being the beat coefficient of the adjacent
$\mathrm{HG}_{n,n}$ and $\mathrm{HG}_{n-1,n}$ modes on a split photodetector.
