<?xml version="1.0"?><ownershipDocument><schemaVersion>X0201</schemaVersion>    <documentType>4</documentType><periodOfReport>2003-10-09</periodOfReport><notSubjectToSection16>0</notSubjectToSection16><issuer><issuerCik>0001054721</issuerCik><issuerName>BSQUARE CORP /WA</issuerName>        <issuerTradingSymbol>BSQR</issuerTradingSymbol></issuer><reportingOwner><reportingOwnerId><rptOwnerCik>0001242316</rptOwnerCik><rptOwnerName>ASP NOGI</rptOwnerName></reportingOwnerId><reportingOwnerAddress><rptOwnerStreet1>3150 - 139TH AVE., SE</rptOwnerStreet1><rptOwnerStreet2>SUITE 500</rptOwnerStreet2><rptOwnerCity>BELLEVUE</rptOwnerCity><rptOwnerState>WA</rptOwnerState><rptOwnerZipCode>98005</rptOwnerZipCode><rptOwnerStateDescription></rptOwnerStateDescription></reportingOwnerAddress><reportingOwnerRelationship><isDirector>0</isDirector><isOfficer>1</isOfficer><isTenPercentOwner>0</isTenPercentOwner><isOther>0</isOther><officerTitle>Vice Pres. Finance, CFO</officerTitle></reportingOwnerRelationship></reportingOwner><derivativeTable><derivativeTransaction><securityTitle><value>Employee Stock Option (Right to Buy)</value></securityTitle><conversionOrExercisePrice><value>1.43</value></conversionOrExercisePrice><transactionDate><value>2003-10-09</value></transactionDate><deemedExecutionDate></deemedExecutionDate><transactionCoding><transactionFormType>4</transactionFormType><transactionCode>A</transactionCode><equitySwapInvolved>0</equitySwapInvolved></transactionCoding><transactionTimeliness><value></value></transactionTimeliness><transactionAmounts><transactionShares><value>50000</value></transactionShares><transactionPricePerShare><value>0</value></transactionPricePerShare><transactionAcquiredDisposedCode><value>A</value></transactionAcquiredDisposedCode></transactionAmounts><exerciseDate><value>2004-01-09</value><footnoteId id="F1" /></exerciseDate><expirationDate><value>2013-10-09</value></expirationDate><underlyingSecurity><underlyingSecurityTitle><value>Common Stock</value></underlyingSecurityTitle><underlyingSecurityShares><value>50000</value></underlyingSecurityShares></underlyingSecurity><postTransactionAmounts><sharesOwnedFollowingTransaction><value>50000</value></sharesOwnedFollowingTransaction></postTransactionAmounts><ownershipNature><directOrIndirectOwnership><value>D</value></directOrIndirectOwnership></ownershipNature></derivativeTransaction></derivativeTable><footnotes><footnote id="F1">The option vests quarterly at a rate of one-eighth of the options per quarter over a 2 year period.</footnote></footnotes><remarks></remarks><ownerSignature><signatureName>Nogi A. Asp</signatureName><signatureDate>2003-10-15</signatureDate></ownerSignature></ownershipDocument>

