According to Hebrews nine, the Jewish tabernacle complex — including its artifacts and rituals — were designed to be "symbolic for the present time" (Heb. 9.9, cf. 1-8). In other words, the physical tabernacle foreshadowed Christian spiritual realties (cf. Heb. 10.1f).
In that same context, the writer mentions "the mercy seat" (9.5). What spiritual truth(s) may we discern from this mercy-seat symbolism?
The Greek Bible (OT and NT) primarily employs the term, hilasterion, in referring to the mercy-seat of Old Testament significance, and that is also the word adopted in Hebrews 9.5.
In Romans 3.25, Paul applies the very same term to Jesus — our hilasterion (i.e., mercy-seat/propitiation) — see  below.
Certainly, Jesus' role as propitiation involves much more than his role as mercy-seat, but failing to perceive at least this aspect of that role (i.e., as mercy-seat) is to overlook a most important facet of human redemption.
In what sense, then, is Jesus our mercy-seat? Consider:
The Jewish temple consisted of two main compartments: (1) the holy place; and (2) the holy of holies. These two chambers were separated by a linen-woven curtain (cf. Ex. 26.31-35).
Through that curtain — within the holy of holies — resided the ark of the covenant (cf. Ex. 25.10f; Lev. 16.12f; Heb. 9.1-5). This was essentially a chest or box containing three artifacts inside (i.e., the golden pot containing manna, Aaron's almond-budding rod, and the stone-tablets of the ten commandments—although, see endnote2 below). The golden "mercy-seat" was placed on top of the ark.
Since the mercy-seat had the same dimensions in length and width as the ark of the covenant, the mercy-seat itself naturally functioned as a lid or cover for the ark, concealing the artifacts inside.
Too, the original Hebrew word used in reference to this mercy-seat is "kapporeth," which generally means, a cover. Hence, the mercy-seat covered the items within the ark of the covenant.
German theologian, Adolph Deissmann (1866-1937), whose work was pivotal in demonstrating that the Greek of the New Testament was the common (koine) language of Greek speakers in that era, observed that the papyri prove that the term hilasterion meant “propitiation;” while the Hebrew, kapporeth, from which hilasterion was translated, simply meant a “lid.” Hence, the Hebrew term identifies its practical function; while the Greek translation replaced that Hebrew concept with hilasterion, which “brings out the [spiritual, AP] purpose of the lid” (The Philology of the Greek Bible, London: Hodder & Stoughton, 1908, cf. pp. 13, 65, 92) — i.e., to blot out sins and bring the people back into fellowship with God.
In that light, it is interesting to note that each of the three artifacts placed inside the ark were reminiscent of Israel's major sins (and of God's compassion for them) as they wandered the wilderness.
The manna reminded Israel of their faithless complaints against Moses and Aaron as they led the people through the wilderness with little to eat (cf. Ex. 16.1f, esp. 11-12; Num. 11.1-9).
Aaron's almond-budding rod reminded them of Korah's rebellion against the divinely-ordained priestly order — the Aaronic sacerdotal system (cf. Num. 16-17).
And the stone-tablets reminded them of their idolatry at the base of Mt. Sinai, at which Moses, with righteous indignation, broke the first set of tablets, requiring the second to be inscribed by God himself (both sets, apparently, were placed within the original ark) —Ex. 32-34.
If each of these artifacts represent the sinfulness of humanity and the condemnation of our race, then Jesus, our mercy-seat, effectively covers the sins of his people from the righteous wrath of God, just as Israel's sins were covered or atoned for symbolically via the mercy-seat.
This meaning is further enhanced by considering the annual ritual involving the Jewish mercy-seat.
Every year, according to levitical law, the Jewish people were required to observe yom kippur (the day of atonement/cover) — cf. Lev. 16.34; Ex. 30.10. On this day, the high priest alone was permitted to enter the holy of holies, once a year.
After offering a bull-sacrifice for himself and his family, the high priest then presented two goats to the Lord on behalf of the people, one of which was to be killed, the other (viz., the scape-goat) was to be set free. The high priest then brought the blood of the slain-goat inside the holy of holies and sprinkled its blood seven times both on and before the mercy-seat (cf. Lev. 16.1-15).
That blood sanctified (and validated) the role of the mercy-seat, and the mercy-seat, in turn, covered the sins of Israel. By this ritual, the high priest symbolically made
"atonement for the holy place, because of the uncleanness of the children of Israel, and because of their transgressions, for all their sins" (Lev. 16.16).
Having done this, the high priest would be permitted to commune with God, who spoke with him from above the mercy-seat in the cloud between the cherubim (cf. Ex. 25.22).
In like manner, though our sins are deserving of sacred punishment (cf. Rm. 6.23), yet God inflicted that punishment on Jesus instead, the sacrificial lamb (cf. Rev. 13.8), and, by taking that blood into the holy of holies (i.e., heaven), Jesus thus covers the sins of those who are in Christ, protecting them from enduring the same wrath-experiencing fate (cf. Heb. 9.12, 24-28; 1 Thess. 1.10; 5.9; Rom. 5.9).
Hence, if we are "in Christ," our mercy-seat, our sins are "covered" and "remembered no more" (Eph. 1.3; Ps. 32.1; Rm. 4.7-8; Jms. 5.20; Heb. 10.16-17).
And because we have obtained mercy, we may have atonement (i.e., at-one-ment) with God; and, as Aaron spoke with God at the mercy-seat, so we may commune with him as often as we need or desire at the throne of grace (i.e., mercy-seat, Heb. 4.16).
In light of all that God has done to become our mercy-seat, should we not, in turn, avail ourselves of the privilege of communing with God at the mercy-seat in worship, service, and prayer with frequent and fervent yearning? Should we not fly swiftly and boldly to our heavenly seat of mercy in times of need and desire?
Indeed! And this is precisely the conclusion of the book of Hebrews — viz., since Jesus is, among other things, our mercy-seat, we ought "therefore" to pray, serve, and attend church-assembly on a frequent basis (cf. Heb. 4.16; 6.18-20; 10.19-25). Such forms of Christian service should not be relegated to the background of our lives, neglected, or in any way taken for granted!
May this reminder of Jesus' role as mercy-seat provoke us all to serve the Lord with diligence in his heavenly sanctuary, the church (cf. Heb. 8.1-2), and allow his ordinances to take precedence over all social and domestic affairs.
"From every stormy wind that blows,
from every swelling tide of woes,
there is a calm, a sure retreat;
'tis found beneath the mercy-seat.
There is a place where Jesus sheds,
the oil of gladness on our heads,
a place than all besides more sweet,
it is the blood-bo't mercy-seat."
 Some dispute the notion that Paul, in Romans 3.25 particularly, was making a symbolic comparison between the mercy-seat of Old Testament significance and Jesus (e.g., Fritzsche, Meyer, Van Hengel, et al.).
However, its predominant use in the Greek Bible as referring to the mercy-seat (cf. Ex. 25:17-22; 31.7; 35.12; 37.6-9; Lev. 16.2, 13-15; Num. 7.89; Heb. 9.5), coupled with the teaching of Hebrews 9.5ff, which demonstrates that the Jewish mercy-seat — and the various rituals and artifacts connected with it — were meant to symbolize the messiah and the messianic order (cf. Heb. 9.1-5, 9-10; 10.1f), both give weight to the symbolic connection between the mercy-seat (hilasterion-propitiation) and Jesus in that passage, at least in a partial sense. I perceive Jesus' role as propitiation as embracing multiple phases or aspects, including his role as high priest, sin-atoning sacrifice, scape-goat, as well as the blood-bought mercy-seat. Romans 3.25 may very well embrace Jesus' propitiatory role in the broad sense, and must not, therefore, exclude his role as mercy-seat.
 It is possible that all three articles (the tablets, pot, and rod) were, spatially, inside the ark at the time when Moses “prepared” it (v.2), as Hebrews 9.4 seems to indicate; but two of the items, the pot and the rod, may have been taken out by the time of Solomon, 500 years later (cf. 1 Kngs. 8.9; 2 Chron. 5.10).
However, as the Greek word, en, is highly elastic, it may rather be employed in an associative sense — i.e., in association with which, etc. Thus, the Hebrews author may have meant that the tablets, pot, and rod were theologically connected with the ark of the covenant, even though only the tablets were located inside the ark.
The Old Testament may support this view, for though the tablets were placed “into” the ark (Ex. 25.16, 21; 40.20), the rod and pot are described as being placed “before” the testimony (Ex. 16.33-34; Num. 17.10). The “testimony” may allude to the tablets themselves (thus, next to the tablets, inside the ark), or, by metonymy, to the ark of the testimony (thus, next to the ark, but not inside it).
Either way, these three items are symbolically “covered” by the golden mercy-seat, and sanctified by the blood.