While it doesn’t positively demand this reading, Romans 8:9-11 furnishes a pattern of thought that is suggestive of the Spirit’s double procession from Father and Son: “But ye are not in the flesh, but in the Spirit, if so be that the Spirit of God dwell in you. Now if any man have not the Spirit of Christ, he is none of his. And if Christ be in you, the body is dead because of sin; but the Spirit is life because of righteousness. But if the Spirit of him that raised up Jesus from the dead dwell in you, he that raised up Christ from the dead shall also quicken your mortal bodies by his Spirit that dwelleth in you.”
Note the alternation: the Spirit is first identified as “of God,” and then “of Christ,” clearly because it is by the Spirit that Christ comes to dwell in believers as their true life amid the death of sin. But then in v. 11, the Spirit is once again identified, not with Jesus, but with “him that raised up Jesus from the dead,” namely, the Father. The Spirit is the Spirit of the Father and the Son, the Spirit from both Father and Son.
And yet even here the Spirit’s relations to Father and Son don’t appear to be perfectly symmetrical; this is also suggestive. The Spirit is the Spirit of Christ because he makes present in us the life that is the Father’s resurrection of the Son; the Spirit’s relation to the Son is inflected by his relation to the Father. We might say, then, with Maximus or Aquinas, that “the Spirit proceeds from the Father and the Son” is equivalent to, “the Spirit proceeds from the Father, through the Son.”