DeleteDuplicates[FullSimplify[Sort@Abs[(Sqrt[8])#],Assumptions->{φ∈Reals,φ>0}]&/@listJLΦ]

JLΦ=octSimplify/@Flatten@prq[prq[octPwrαL,TL,1],octPwrαLsw,cpL];
Length@%
JLΦRnd=rndOct/@%%;
listJLΦ=FullSimplify[oct2List[#]&/@%%%,Assumptions->{φ∈Reals,φ>0}];
listJLΦRnd=oct2List/@%%;
(**)
hulls3DPerms["listJLΦ",False,,1]

tally3=Sort@Tally[{tmp=Sort@rndMat[(Sqrt[8])Abs@#/.φ->(1/2) (-1.+(Sqrt[5]))];
octSym[tmp]/.φ->1/φ,tmp}&/@listJLΦ[[All,;;4]],#1[[2]]==#2[[2]]&];
{Column[{Row@{#[[1,1]],"/(Sqrt[8])"},#[[1,2]]/(Sqrt[8])},Center],#[[2]]}&/@tally3;
%//MatrixForm

diminishedJΦInv3=Select[listJLΦ,
FullSimplify[Sort@Abs[(Sqrt[8])#],Assumptions->{φ∈Reals,φ>0}]=={0,0,0,0,0,0,2,2}||
FullSimplify[Sort@Abs[(Sqrt[8])#],Assumptions->{φ∈Reals,φ>0}]=={0,0,0,0,0,1,(1/(φ^2)),(φ^2)}&];
Length@%
hulls3DPerms["diminishedJΦInv3",False,,1]

diminishedJΦ3=Select[listJLΦ,!MemberQ[diminishedJΦInv3,#]&];
Length@%
hulls3DPerms["diminishedJΦ3",False,,1]

diminishedJΦInv4=Select[listJLΦ,
FullSimplify[Sort@Abs[(Sqrt[8])#],Assumptions->{φ∈Reals,φ>0}]=={0,0,0,0,0,0,2,2}||
FullSimplify[Sort@Abs[(Sqrt[8])#],Assumptions->{φ∈Reals,φ>0}]=={0,0,0,0,0,(Sqrt[5]),(1/φ),φ}&];
Length@%
hulls3DPerms["diminishedJΦInv4",False,,1]

diminishedJΦ4=Select[listJLΦ,!MemberQ[diminishedJΦInv4,#]&];
Length@%
hulls3DPerms["diminishedJΦ4",False,,1]

diminishedJΦInv5=Select[listJLΦ,
(*)FullSimplify[Sort@Abs[(Sqrt[8])#],Assumptions->{φ∈Reals,φ>0}]=={0,0,0,0,0,0,2,2}||**)
FullSimplify[Sort@Abs[(Sqrt[8])#],Assumptions->{φ∈Reals,φ>0}]=={0,0,0,0,1,2,(1/φ),φ}&];
Length@%
hulls3DPerms["diminishedJΦInv5",False,,1]

diminishedJΦ5=Select[listJLΦ,!MemberQ[diminishedJΦInv5,#]&];
Length@%
hulls3DPerms["diminishedJΦ5",False,,1]

diminishedJΦInv6=Select[listJLΦ,
(*)FullSimplify[Sort@Abs[(Sqrt[8])#],Assumptions->{φ∈Reals,φ>0}]=={0,0,0,0,0,0,2,2}||**)
FullSimplify[Sort@Abs[(Sqrt[8])#],Assumptions->{φ∈Reals,φ>0}]=={0,0,0,0,1,1,1,(Sqrt[5])}||
FullSimplify[Sort@Abs[(Sqrt[8])#],Assumptions->{φ∈Reals,φ>0}]=={0,0,0,0,(1/φ),(1/φ),(1/φ),(φ^2)}||
FullSimplify[Sort@Abs[(Sqrt[8])#],Assumptions->{φ∈Reals,φ>0}]=={0,0,0,0,(1/(φ^2)),φ,φ,φ}&];
Length@%
hulls3DPerms["diminishedJΦInv6",False,,1]

diminishedJΦ6=Select[listJLΦ,!MemberQ[diminishedJΦInv6,#]&];
Length@%
hulls3DPerms["diminishedJΦ6",False,,1]