I am confused by a couple of theorems of Weibel's "Introduction to Homological Algebra". In there a result for left hyper-derived functors is seemingly stronger than its dual for right hyper-derived functors.
To get left hyper-derived functors $\mathbb{L}_iF(A_*)$ one considers a projective Cartan-Eilenberg resolution of $A_*$ as an upper half-plane double complex $P$ and takes the homology of the total complex $\text{Tot}^{\bigoplus}(F(P))$. Then we get 2 spectral sequences:
$$ ^{II}E^2_{pq}=(L_pF)(H_q(A_*))\Longrightarrow \mathbb{L}_{p+q}F(A_*)$$
converging in all cases, and
$$ ^{I}E^2_{pq}=H_p(L_qF(A_*))\Longrightarrow \mathbb{L}_{p+q}F(A_*)$$ converging if $A_*$ is bounded below.
Now to get right hyper-derived functors $\mathbb{R}^iF(A^*)$ one considers the injective Cartan-Eilenberg resolution of $A^*$ as an upper half-plane double cochain complex $I$ and take the cohomology of the total complex $\text{Tot}^{\prod}(F(I))$. Then we also get 2 spectral sequences:
$$ ^{II}E_2^{pq}=(R^pF)(H^q(A^*))\Longrightarrow \mathbb{R}^{p+q}F(A_*)$$
weakly convergent, and
$$ ^{I}E_2^{pq}=H^p(R^qF(A^*))\Longrightarrow \mathbb{R}_{p+q}F(A^*)$$ converging if $A_*$ is bounded below.
It seems that the two pairs of spectral sequences for homology and cohomology are dual with respect to rotating the double complex by $180^\circ$ around the origin, and not with respect to inverting the arrows. This makes me think that there should be 4 natural spectral sequences for each case. For example, for homology these would correspond to double complexes truncated below some horizontal and to the left of some vertical. The only problem I can see with writing these sequences down is the indexing: for filtrations on double chain complexes people agreed to that $F_p\subset F_{p+1}$, while for cochain complexes we agreed that $F^p\subset F^{p-1}$, and in this notation the two "missing" spectral sequences are simply inconvenient to write down.
It also seems that these 2 sequences are encountered when we switch to the opposite category. Under this switch my injective Cartan-Eilenberg resolution becomes projective, right hyper-derived functors become left hyper-derived functors, and I can write down the $^II E$ spectral sequence which always converges. Then I finally return to the original category with a converging spectral sequence not predicted by the theorem.
So the questions are:
Am I right that there are 4 "natural" spectral sequences for double (co)chain complexes and that we only choose to write down two of them becomes of our choice in notation. If so, then what's the story behind this choice? And are there any other situations in which this choice results in asymmetric results for homology and cohomology?