To my knowledge, there are no Julia packages with good support for directional and orientational statistics. Directional statistics is used in biology (especially structural biology), crystallography, astronomy, and various other physics applications including geophysics.
Distributions.jl implements VonMises
for circular variables (angles) and VonMisesFisher
for spherical variables (unit vectors), but that’s it. No orientational distributions are implemented.
From the discussion on slack, it seems like a package extending Distributions.jl and Statistics.jl to handle these kinds of distributions could be useful, and I’d like to 1) gauge interest in such a package, 2) present a rough design proposal for feedback, and 3) find potential contributors since I only need a subset of these features for my research.
Goals:
 Implement common distributions/statistics in directional/orientational statistics
 Compatible with common AD frameworks
 Efficient fitting (where possible)
 Usable within PPLs like Soss, Turing, and Gen
 Realize directional statistics as a special case of statistics on Riemannian manifolds by using Manifolds.jl’s manifolds and interface
 Stress test for Manifolds.jl distributions interface (mostly written by @mateuszbaran; see also Simplifying Distributions type hierarchy)
Covered Manifolds
These are the manifolds we will consider for defining distributions. Some are already implemented in Manifolds.jl.
Directional
Circle
Manifolds.Sphere

Hemisphere
? (for points that represent axes) Torus
Orientational

Manifolds.Stiefel
(WIP by @kellertuer) Manifolds.Rotations
Distributions
Some of these are sufficiently general that they might make their way into Manifolds.jl or a package like ManifoldsDistributions.jl.
Generic (have a Manifold
type for specialization)
Dirac
Uniform
 Normal analogs

ProjectedTangentNormal
(normal in tangent space projected to manifold) 
RiemannianNormal
(normal in manifold) 
IsotropicDiffusion
(solution to heat equation)

Mixture
Directional

Circle
Wrapped{<:ContinuousUnivariateDistribution}
VonMises

Sphere
VonMisesFisher
Kent

Hemisphere
Watson

Torus

CircleProductDistribution
(product of NCircle
s) e.g.
NvariateVonMises
 e.g.

Orientational

Stiefel

TupleDistribution{Stiefel}
? (SphereProductDistribution
restricted to orthonormal) e.g.
MatrixVonMisesFisher
 e.g.


Rotations
 Comprise from
Stiefel
/Sphere
 Comprise from
Statistics
 mean resultant length
 others…