A set of paragraphs of 4 lines to manage with AWK

awk -F[][] -vOFS= '++i==1 {a=$2} i==2 {b=$2} i==3 {$2="[" b "]"} i==4 {$2="[" a "]"} !NF {i=0} 1' input.txt

With square brackets as the field separators, your replacement sources/targets are in $2.

We increment i on each line, and reset it to zero between paragraphs. The value of i (1 though 4) tells us what to do with $2.

$ cat tst.awk
match($0,/\[.*]/) {
    idx = (NR - 1) % 5 + 1
    sect[idx] = substr($0,RSTART,RLENGTH)
    if ( idx == 3 ) {
        $0 = $1 OFS sect[2] OFS $NF
    }
    else if ( idx == 4 ) {
        $0 = $1 OFS sect[1] OFS $NF
    }
}
{ print }

$ awk -f tst.awk file
A1 [A3 A4 A5] A2
B1 [B3 B4 B5] B2
C1 [B3 B4 B5] C2
D1 [A3 A4 A5] D2

E1 [E3 E4 E5] E2
F1 [F3 F4 F5] F2
G1 [F3 F4 F5] G2
H1 [E3 E4 E5] H2

The above does string replacement so it'll work even if the sections inside brackets contain regexp metachars or backreferences.