Defining Rolfsen Screw

twist={r{x,Sin[3/2 x],Cos[3/2 x]}+(1-r){2π/3 -x,Sin[3/2(2π/3-x)],Cos[3/2(2π/3-x)]}+{2/3π n,0,0},r{x,-Sin[3/2 x],-Cos[3/2 x]}+(1-r){2π/3 -x,-Sin[3/2(2π/3-x)],-Cos[3/2(2π/3-x)]}+{2/3π n,0,0}}

ParametricPlot3D[twist/.n->0,{x,0,2π/3},{r,0,1}]

trefoiltwist=Flatten[Table[twist,{n,0,2}],1]
ParametricPlot3D[trefoiltwist,{x,0,2π/3},{r,0,1},Exclusions->None]

R[{x_,y_,z_},θ_]:={x+θ/3,y Cos[θ/2]+z Sin[θ/2],-y Sin[θ/2]+z Cos[θ/2]}

R[#,π/2]&/@trefoiltwist//Simplify
ParametricPlot3D[R[#,π/2]&/@trefoiltwist//Simplify,{x,0,2π/3},{r,0,1}]

Different Torus Projection Attempt

onetwist
ParametricPlot3D[onetwist/.n->0,{x,0,2π/3},{r,0,1}]

Theta[{x_,y_,z_}]:=ArcTan[y,z]
Phi[{x_,y_,z_}]:=x

AnotherTorus[leaf_]:=Module[{θ=Theta[leaf],ϕ=Phi[leaf]},
{(2+r Cos[θ])Cos[ϕ],(2+r Cos[θ])Sin[ϕ],r Sin[θ]}//Simplify]

ParametricPlot3D[AnotherTorus/@trefoiltwist,{x,0,2π/3},{r,0,1}]

Show[
ParametricPlot3D[{(2+Cos[θ])Cos[ϕ],(2+Cos[θ])Sin[ϕ],Sin[θ]},{θ,0,2π},{ϕ,0,2π},PlotStyle->Directive[Opacity[0.2]],Mesh->None,PlotPoints->100],ParametricPlot3D[AnotherTorus/@trefoiltwist,{x,0,2π/3},{r,0,1},Mesh->None],
ParametricPlot3D[RotationMatrix[π/3,{0,0,1}].{(2-Cos[3t])Cos[2t],(2-Cos[3t])Sin[2t],Sin[3t]},{t,0,2π},PlotStyle->{Thick}]

]

Show[
ParametricPlot3D[{(2+Cos[θ])Cos[ϕ],(2+Cos[θ])Sin[ϕ],Sin[θ]},{θ,0,2π},{ϕ,0,2π},PlotStyle->Directive[Opacity[0.2]],Mesh->None,PlotPoints->100],ParametricPlot3D[AnotherTorus/@((R[#,Pi]&/@trefoiltwist)//Simplify)//Simplify,{x,0,2π/3},{r,0,1},Mesh->None],
ParametricPlot3D[RotationMatrix[π/3,{0,0,1}].{(2-Cos[3t])Cos[2t],(2-Cos[3t])Sin[2t],Sin[3t]},{t,0,2π},PlotStyle->{Thick}]

]

trefoiltwist

R[#,0]&/@trefoiltwist

AnotherTorus/@((R[#,0]&/@trefoiltwist)//Simplify)//Simplify

Projecting to Torus

onetwist=twist[[1]]/.n->1
zRot[θ_]:=RotationMatrix[θ,{0,0,1}]
TorusProject[leaf_]:=Module[
{θ={1,0,0}.leaf},
zRot[θ].({0,1,1}*leaf)+{Cos[θ],Sin[θ],0}
]

ParametricPlot3D[zRot[π/2].({0,1,1}*onetwist)+{Cos[{1,0,0}.onetwist],Sin[{1,0,0}.onetwist],0},{x,0,2π/3},{r,0,1}]

trefoiltwist

trefoilscrew=((Table[Map[RotateFunction,trefoiltwist],{n,0,2}])//Simplify)

ParametricPlot3D[trefoilscrew,{x,0,2π/3},{r,0,1}]

Badlands

Manipulate[
Show[ParametricPlot3D[RotationMatrix[θ,{0,0,1}].Table[twist,{n,0,0}][[1,1]],{x,0,2π/3},{r,0,1}],Graphics3D[Arrow[{{0,0,0},twist[[1]]/.{r->0.5,x->0.5,n->0}}]]],{θ,0,2π}]

twist[[1]]/.{r->0.5,x->0.5,n->0}

Graphics3D[Arrow[{{0,0,0},twist[[1,1]]/.{r->0.5,x->0.5}}]]

Table[twist,{n,0,0}][[1,1]]

ParametricPlot3D[Table[twist,{n,0,2}],{x,0,2π/3},{r,0.8,1}]

ParametricPlot3D[Table[twist,{n,0,2}],{x,0,2π/3},{r,0,1}]

R[{x_,y_,z_},θ_]:={x+θ/3,y Cos[θ/2]+z Sin[θ/2],-y Sin[θ/2]+z Cos[θ/2]}

ParametricPlot3D[Table[q@R[#,0]&/@twist,{n,0,3}],{x,0,2π/3},{r,0,1}]

q[{x_,y_,z_},r_:1,R_:2]:={R +r Sin[x],R +r Cos[x]y,z}

Torus[r_:1,R_:2]:={(R+r Cos[θ])Cos[ϕ],(R+r Cos[θ])Sin[ϕ],r Sin[θ]}

ParametricPlot3D[Torus[],{θ,0,2Pi},{ϕ,0,2Pi}]

θ=rx+(1-r)(2(π/3)-x)+2/3 π n
torustwist={RotationMatrix[θ,{0,0,1}].(r{x,Sin[3/2 x],Cos[3/2 x]}+(1-r){2π/3 -x,Sin[3/2(2π/3-x)],Cos[3/2(2π/3-x)]}-{θ,0,0}+{2/3π n,0,0}+2{Cos[θ],Sin[θ],0}),RotationMatrix[θ,{0,0,1}].(r{x,-Sin[3/2 x],-Cos[3/2 x]}+(1-r){2π/3 -x,-Sin[3/2(2π/3-x)],-Cos[3/2(2π/3-x)]})+{2/3π n,0,0}-{θ,0,0}+2{Cos[θ],Sin[θ],0}}

ParametricPlot3D[Table[torustwist,{n,0,2}],{x,0,2π/3},{r,0,1}]

CoordinateTransform["Cartesian"->"Cylindrical",{1,1,1}]