The text almost gives the answer.
The large animals, heifer, goat and ram were to be cut in two, and the two birds left uncut. Abram then placed the halved animals on either side of a short path - the two doves were placed one on each side of the path as well (uncut).
Thus, the birds were placed opposite each other. Gill summarizes this as follows:
but the birds divided he not; but laid them one against another, as
the pieces were laid; so the birds used in sacrifice under the law
were not to be divided, Leviticus 1:17; which may signify, that when
the people of the Jews, in the latter day, are converted, and brought
together into their own land, when they will better answer the
character of turtles and doves than they ever did, will be no more
divided and separated from each other.
The pulpit commentary offers a similar interpretation:
but the birds divided he not. So afterwards in the Mosaic legislation (Leviticus 1:7). Wordsworth detects in the non-dividing of
the birds an emblem of "the Holy Spirit, the Spirit of peace and love;
which is a Spirit of unity, and of "Christ's human spirit, which was
not divisible." Kalisch, with more probability, recognizes as the
reason of their not being divided the fact that such division was not
required, both fowls being regarded as one part of the sacrifice only,
and each, as the half, being placed opposite the other.