diff --git a/README.md b/README.md
index fd7ae016..486bd2ff 100644
--- a/README.md
+++ b/README.md
@@ -649,6 +649,14 @@ await wtnsCalculate(input, wasmFile, wtns, {memorySize: 0});
 
 
 
+### G2 point ordering in the Solidity verifier
+
+The exported Groth16 verifier expects the proof's `B` component in EIP-197
+order, which is the reverse of `proof.json`'s. `zkey export soliditycalldata`
+performs that swap for you, so its output can be passed to `verifyProof`
+directly. If you build the call by hand, see the note in the generated
+verifier above its `// B` block.
+
 ## Further resources
 - [Announcing the Perpetual Powers of Tau Ceremony to benefit all zk-SNARK projects](https://medium.com/coinmonks/announcing-the-perpetual-powers-of-tau-ceremony-to-benefit-all-zk-snark-projects-c3da86af8377)
 - [Scalable Multi-party Computation for zk-SNARK Parameters in
diff --git a/cli.js b/cli.js
index 4482f3e4..c3c801a6 100755
--- a/cli.js
+++ b/cli.js
@@ -243,13 +243,26 @@ const commands = [
     },
     {
         cmd: "zkey export solidityverifier [circuit_final.zkey] [verifier.sol]",
-        description: "Creates a verifier in solidity",
+        description: "Creates a verifier in solidity. Expects _pB in EIP-197 order; see soliditycalldata",
+        longDescription: "Creates a verifier in solidity.\n\n" +
+            "The generated contract's verifyProof expects the proof's B component\n" +
+            "in EIP-197 order ([x_im, x_re, y_im, y_re]) -- the reverse of\n" +
+            "proof.json's; see the note above the contract's `// B` block.  Use\n" +
+            "'zkey export soliditycalldata' to produce arguments in that order.",
         alias: ["zkesv", "generateverifier -vk|verificationkey -v|verifier"],
         action: zkeyExportSolidityVerifier
     },
     {
         cmd: "zkey export soliditycalldata [public.json] [proof.json]",
-        description: "Generates call parameters ready to be called.",
+        description: "Generates call parameters ready to be called (proof's B already in EIP-197 order)",
+        longDescription: "Generates call parameters ready to be called.\n\n" +
+            "The proof's B component is emitted in EIP-197 order\n" +
+            "([x_im, x_re, y_im, y_re]), which is the reverse of proof.json's and\n" +
+            "is what the exported verifier expects -- so this output can be\n" +
+            "passed to verifyProof directly.  Building the call by hand from\n" +
+            "proof.json without swapping each G2 coordinate pair yields an\n" +
+            "off-curve point, and verifyProof then returns false for a proof\n" +
+            "that verifies fine off-chain.",
         alias: ["zkesc", "generatecall -pub|public -p|proof"],
         action: zkeyExportSolidityCalldata
     },
diff --git a/smart_contract_tests/test/smart_contracts.test.js b/smart_contract_tests/test/smart_contracts.test.js
index d24f8bcf..5f6467ae 100644
--- a/smart_contract_tests/test/smart_contracts.test.js
+++ b/smart_contract_tests/test/smart_contracts.test.js
@@ -39,6 +39,13 @@ describe("Smart contracts test suite", function () {
         )).to.be.equal(true);
     });
 
+    it("Groth16 exportSolidityCallData output verifies on-chain", async () => {
+        expect(await groth16VerifyViaExportedCallData(
+            path.join("../test", "groth16", "circuit.r1cs"),
+            path.join("../test", "groth16", "witness.wtns")
+        )).to.be.equal(true);
+    });
+
     it("Groth16 smart contract 1 aliased input", async () => {
         expect(
             await groth16VerifyAliased(
@@ -92,6 +99,10 @@ describe("Smart contracts test suite", function () {
         const { proof: proof, publicSignals: publicInputs } = await snarkjs.groth16.prove(zkeyFilename, wtnsFilename);
 
         const proofA = [proof.pi_a[0], proof.pi_a[1]];
+        // G2 coordinates are swapped to EIP-197 order here; the verifier
+        // expects _pB im-first.  exportSolidityCallData does the same swap --
+        // see the groth16 exportSolidityCallData test below, which pins that
+        // exporter and verifier agree.
         const proofB = [[proof.pi_b[0][1], proof.pi_b[0][0]], [proof.pi_b[1][1], proof.pi_b[1][0]]];
         const proofC = [proof.pi_c[0], proof.pi_c[1]];
 
@@ -109,6 +120,34 @@ describe("Smart contracts test suite", function () {
         return await verifierContract.verifyProof(proofA, proofB, proofC, publicInputs);
     }
 
+    // Pins the contract between `zkey export soliditycalldata` and the exported
+    // verifier.  Nothing else covers it: every other test builds the calldata by
+    // hand, so both could drift from each other and stay green.  The G2 ordering
+    // convention -- verification-key constants im-first inside the contract, the
+    // proof's B swapped by the caller -- is only correct if these two agree, and
+    // that agreement is an ABI: verifiers are deployed immutably, so changing
+    // either side desynchronises every deployment that already exists.
+    async function groth16VerifyViaExportedCallData(r1csFilename, wtnsFilename) {
+        const solidityVerifierFilename = path.join("contracts", "groth16.sol");
+        const zkeyFilename = { type: "mem" };
+
+        await snarkjs.zKey.newZKey(r1csFilename, ptauFilename, zkeyFilename);
+        const { proof, publicSignals } = await snarkjs.groth16.prove(zkeyFilename, wtnsFilename);
+
+        const verifierCode = await snarkjs.zKey.exportSolidityVerifier(zkeyFilename, templates);
+        fs.writeFileSync(solidityVerifierFilename, verifierCode, "utf-8");
+        await run("compile");
+
+        const VerifierFactory = await ethers.getContractFactory("Groth16Verifier");
+        verifierContract = await VerifierFactory.deploy();
+
+        // Take the arguments exactly as the CLI would emit them.
+        const callData = await snarkjs.groth16.exportSolidityCallData(proof, publicSignals);
+        const [a, b, c, inputs] = JSON.parse(`[${callData}]`);
+
+        return await verifierContract.verifyProof(a, b, c, inputs);
+    }
+
     async function groth16VerifyAliased(r1csFilename, wtnsFilename) {
         const solidityVerifierFilename = path.join("contracts", "groth16.sol");
 
diff --git a/templates/verifier_groth16.sol.ejs b/templates/verifier_groth16.sol.ejs
index fda549e3..43d1bb14 100644
--- a/templates/verifier_groth16.sol.ejs
+++ b/templates/verifier_groth16.sol.ejs
@@ -105,6 +105,24 @@ contract Groth16Verifier {
                 mstore(add(_pPairing, 32), mod(sub(q, calldataload(add(pA, 32))), q))
 
                 // B
+                //
+                // NOTE ON G2 ORDERING.  `_pB` is expected to ALREADY be in
+                // EIP-197 order, i.e. [x_im, x_re, y_im, y_re], and is passed
+                // through unchanged here.  The verification-key constants
+                // above (betax1/betax2/...) are likewise emitted im-first.
+                //
+                // The swap therefore happens on the CALLER side, not in this
+                // contract.  `snarkjs zkey export soliditycalldata` performs
+                // it for you (see src/groth16_exportsoliditycalldata.js).  If
+                // you build calldata by hand from a proof.json, you must swap
+                // each G2 coordinate pair yourself:
+                //
+                //     B = [[pi_b[0][1], pi_b[0][0]],
+                //          [pi_b[1][1], pi_b[1][0]]]
+                //
+                // Passing proof.json's natural order straight through yields
+                // an off-curve point; the pairing precompile then fails and
+                // verifyProof returns false for an otherwise valid proof.
                 mstore(add(_pPairing, 64), calldataload(pB))
                 mstore(add(_pPairing, 96), calldataload(add(pB, 32)))
                 mstore(add(_pPairing, 128), calldataload(add(pB, 64)))