Try this:

df.sort_values( ts ).assign(sortkey=df.groupby( action ).cumcount()).sort_values([ sortkey , action ])

Output:

                         ts action      name  sortkey
0   2023-06-05 16:51:27.561    ACQ  location        0
3   2023-06-05 16:51:27.571    REL  location        0
1   2023-06-05 16:51:27.564    ACQ  location        1
4   2023-06-05 16:51:27.573    REL  location        1
2   2023-06-05 16:51:27.567    ACQ  location        2
5   2023-06-05 16:51:27.587    REL  location        2
6   2023-06-05 16:51:28.559    ACQ  location        3
9   2023-06-05 16:51:28.566    REL  location        3
7   2023-06-05 16:51:28.561    ACQ  location        4
10  2023-06-05 16:51:28.569    REL  location        4
8   2023-06-05 16:51:28.563    ACQ  location        5
11  2023-06-05 16:51:28.575    REL  location        5

If it is always guaranteed that there are the same number of ACQ and REL, and that they are respectively in correct order (i.e., the n-th ACQ should always be paired with the n-th REL), then the solution is to split the original list into two first, by its second element, then loop to output one from each list.

Code:

import pandas as pd

data = [
    [ 2023-06-05 16:51:27.561 ,  ACQ ,  location ],
    [ 2023-06-05 16:51:27.564 ,  ACQ ,  location ],
    [ 2023-06-05 16:51:27.567 ,  ACQ ,  location ],
    [ 2023-06-05 16:51:27.571 ,  REL ,  location ],
    [ 2023-06-05 16:51:27.573 ,  REL ,  location ],
    [ 2023-06-05 16:51:27.587 ,  REL ,  location ],
    [ 2023-06-05 16:51:28.559 ,  ACQ ,  location ],
    [ 2023-06-05 16:51:28.561 ,  ACQ ,  location ],
    [ 2023-06-05 16:51:28.563 ,  ACQ ,  location ],
    [ 2023-06-05 16:51:28.566 ,  REL ,  location ],
    [ 2023-06-05 16:51:28.569 ,  REL ,  location ],
    [ 2023-06-05 16:51:28.575 ,  REL ,  location ]
]

df = pd.DataFrame(data, columns=[ ts ,  action ,  name ])
# print(df)

data_acq = []
data_rel = []

for index, row in df.iterrows():
    if row[ action ] ==  ACQ :
        data_acq.append(row)
    elif row[ action ] ==  REL :
        data_rel.append(row)
assert len(data_acq) == len(data_rel)

df_new = pd.DataFrame([], columns=[ ts ,  action ,  name ])

for j in range(len(data_acq)):
    df_new = pd.concat([
        df_new,
        pd.DataFrame([
            data_acq[j], data_rel[j]
        ])
    ])
print(df_new)

Output:

                         ts action      name
0   2023-06-05 16:51:27.561    ACQ  location
3   2023-06-05 16:51:27.571    REL  location
1   2023-06-05 16:51:27.564    ACQ  location
4   2023-06-05 16:51:27.573    REL  location
2   2023-06-05 16:51:27.567    ACQ  location
5   2023-06-05 16:51:27.587    REL  location
6   2023-06-05 16:51:28.559    ACQ  location
9   2023-06-05 16:51:28.566    REL  location
7   2023-06-05 16:51:28.561    ACQ  location
10  2023-06-05 16:51:28.569    REL  location
8   2023-06-05 16:51:28.563    ACQ  location
11  2023-06-05 16:51:28.575    REL  location

I don t know if this can be translated into pandas equivalents, but in plain Python with accumulate from itertools, you can compute a list of offsets for the REL that indicate how far back the corresponding ACQ is located. Then sort the indices according to the adjusted positions (ACQ remaining at their original position and REL offset back to the ACQ s position) to get the indexes in the new order:

from itertools import accumulate

isREL   = (int(x=="REL") for _,x,_ in data)
offsets = accumulate(isREL,lambda a,b:(a or 1)*b-2*b)
order   = (i for i,_ in sorted(enumerate(offsets),key=sum))
data    = [data[i] for i in order]

print(*data,sep="
")

[ 2023-06-05 16:51:27.561 ,  ACQ ,  location ]
[ 2023-06-05 16:51:27.587 ,  REL ,  location ]
[ 2023-06-05 16:51:27.564 ,  ACQ ,  location ]
[ 2023-06-05 16:51:27.573 ,  REL ,  location ]
[ 2023-06-05 16:51:27.567 ,  ACQ ,  location ]
[ 2023-06-05 16:51:27.571 ,  REL ,  location ]
[ 2023-06-05 16:51:28.559 ,  ACQ ,  location ]
[ 2023-06-05 16:51:28.575 ,  REL ,  location ]
[ 2023-06-05 16:51:28.561 ,  ACQ ,  location ]
[ 2023-06-05 16:51:28.569 ,  REL ,  location ]
[ 2023-06-05 16:51:28.563 ,  ACQ ,  location ]
[ 2023-06-05 16:51:28.566 ,  REL ,  location ]